Virginia Poe. Subverted in that 1) They knew exactly what she had, and 2) it was a foregone conclusion (though some say that if Edgar hadn't been such a starving artist, he might have been able to pay well enough to help her recover). As it is, she lasted years. This showed up a lot in his works as a result.
The trope is very common among female Catholic servants of God, venerables, blesseds, saints and visionaries. Some of them are:
Sister Agnes Katsuko Sasagawa, a Japanese nun from the Yuzawadai prefecture in Akita, who would become the main witness in the Marian Apparition known as Our Lady of Akita. Ever since a poorly-performed operation, Sister Sasagawa had many health problems that got her paralyzed and later deaf; her health apparently improved after drinking some holy water from Lourdes, and her deafness was finally healed in The Eighties. Additionally, Sister Sasagawa also developed stigmata (allegedly) at the same time the local Mary statue cried Tears of Blood.
Therese Neumann. As a child she sustained quite the injuries when she took a bad fall from a stool, years later she was rendered blind and also was confined to her bed due to her injured back and limbs. Allegedly recovered her sight and later her mobility due to the intervention of her favorite saint, Saint Therese of Lisieux; later she got appendicitis but was miraculously cured right before being operated. OTOH, the then-healthy Therese would develop stigmata, and later led a very hard life during the Third Reich and World War II, with the Gestapo keeping an eye on her and attacking her friends and family since she openly critisized the Nazi regime but was so popular among German people that they couldn't attack her directly.
Saint Lidwina of Schiedam, a Dutch minor noblewoman who at age 15 was paralyzed and confined to a bed after an accident while ice skating, then became a famous holy woman and faith healer. It's said that, among other things, "she became paralyzed except for her left hand and that great pieces of her body fell off, and that blood poured from her mouth, ears, and nose", and said body parts were supposedly stored and then lost. (And sadly, it's also known that she was sexually abused to different degrees, either by soldiers or local priests). It's believed that Lidwina was likely affected by multiple sclerosis, rather than mere damage taken by her fall.
Venerable Antonietta "Nennolina" Meo, who had bone cancer and lost a leg for it before dying (but allegedly saw it as a sacrifice to God) and wrote many letters to Jesus Christ before passing away.
Marie Francoise Therese Guerin aka Saint Therese of Lisieux, who also was a Book Worm. She wrote her very mystical autobiography, Story of a Soul (which made her worthy of being a "Doctor of the Church"(note a special title among Christian saints given *only* to those who led a very saintly life by Christian standards, were NOT Martyrs of the Faith, AND massively contributed to doctrine or theology in one way or another), while afflicted with tuberculosis that would kill her at age 24.
Likewise Blessed Cecilia Eusepi, who wrote her book Story of a Clown following Therese's example, and died at age 18.
And then... Yuko Goto, although some of that is her own fault. She was diagnosed of Idiopathic thrombocytopenic purpura as a teen (which eventually developed into Lupus) and could have taken a less stressful job instead, but she took the work of a seiyuu and didn't take regular blood counts to track her condition. It was reported by 2009 she already needs OTC painkillers to function, in 2011 she could no longer climb stairs (and has to be carried by her manager), and in summer 2012 she was hospitalized several months for inflammation in multiple organs. As of October 2012 she may be out but still in immunosuppresive therapy...
On the other side of the Pacific, Carrie Savage suffers from muscular dystrophy. Anime Detour, a convention from Minnesota, held a fundraiser to cover her hospital expenses in 2013.
Anna Rogel (1751–1784), a Finnish preacheress. She was paralyzed for life by most likely poliomyelitis in 1770 and was bed-ridden for the rest of her life. She became a famous preacher and was one of the founders of the spiritual movement in southwestern Finland.
Eva Duarte de Perón aka "Evita" is an older version, as she died of cancer at age 33.
The Radium Girls, girls who painted glow in the dark watches with radium paint in a factory, some starting as young as 15. Upon realizing the they had gotten severe radiation poisoning, five of them sued their employers and became a media sensation in the 1920s.
Mangaka Minami Ohzaki from Zetsuai1989, who had to completely halt the manga for years because of her health.
Another mangaka, Kyouko Okazaki, became a horror version of this after being hit by a car in 1996. The poor woman had such injuries that she was left quadriplegic AND mute, and she's still in rehab after almost 15 years. Her assistant Moyoco Anno (wife of thatAnno) had to finish her horror manga Helter Skelter.
Esther Earl, an influential and inspiring Nerdfighter was one of these. She suffered from cancer for four years and died earlier this year (2010) at age 16. As you can see from her videos, she was as sweet and hopeful a person as any fictional example.
Kazuya Minekura, the author of Saiyuki. First she had an undiclosed illness that made her have to go through an hysterectomy, and later had a tumor removed from her jaw.
Sadako Sasaki was two years old when an atom bomb was dropped one mile from her home in Hiroshima. Her story was memorialized in the book Sadako and the Thousand Paper Cranes, which was inspired by the belief that if a person managed to fold a thousand paper cranes, they would be granted a wish from the gods. She died at the age of 12.
Lili Boulanger, composer, aged 24. Invoked by her artistic output, which dwells a fair bit on the whole "We Are as Mayflies" thing.
The infamous KayceeNicole hoax was based on one of these... who actually never existed.
While not as common as the Ill Girls, the Catholic Church also has Ill Boys among their venerables, blesseds and saints. Many of them suffered of tuberculosis:
The aforementioned Blessed Francisco Marto, Fatima visionary. Died of Spanish influenza at age 11, along with his little sister Jacinta.
Francesco Possenti, alias Saint Gabriel Of Our Lady Of Sorrows. Had already a bad health as a youngster, was close to die at least twice before turning 18 (and entered the Passionist Order as thanks to the Virgin Mary for his healing), and died of tuberculosis at age 24.
Blessed Ceferino Namuncurá, Argentinian Country Mouse who worked hard to become a Salesian priest despite having a delicate health. He fell victim to tuberculosis after a trip to Rome, and died at age 19.
Blessed Edmund Bojanowski, Polish layman who almost died as a kid and later developed tuberculosis, which didn't let him apply to priesthood. Instead, he dedicated himself to charity and caretaking.
Yukito Kishiro, author of Battle Angel Alita (also known as Gunnm), had to finish it much earlier than he thought because of his bad health and his unwillingness to have an Author Existence Failure happen. He got better, though, and restarted the manga once he was back.
Formula One driver Gerhard Berger had a brief time as one in 1997, having to take at least three races (equalling to several weeks) out due to his health. For worse, that was the time when his father died too, causing him an Heroic BSOD.
Yoshiki Hayashi qualifies. As a child he was ill from his mother's difficult labor and crippling asthma. Eventually he got better but then he's had everything from "neurocirculatory asthenia" to mental illness to physical damage from his intense drumming to hyperthyroidism...
For Tear Jerker value, his son Patrick was born prematurely... and died two days after his birth. And his and Jackie's first-born child and oldest daughter, Arabella, was still-born. To put things in perspective, this means that John and Jackie lost both their oldest and youngest children either right before or directly after their births.
Edward VI, the last Tudor King of England, was horribly ill with either tuberculosis or acute kidney failure for the last few months of his short life, and died at age sixteen.
Similarly, his uncle Arthur Tudor died when he was around 15, due to either TB, diabetes or some kind of sweating sickness. This was actually a big Spanner in the Works for his father, Henry VII, since Arthur was the eldest son and his other son, Henry VIII, was never intended to be king.
Tsarevich Alexei Romanov, heir to the Russian throne, was plagued throughout his short life by hemophilia, coming close to death many times. Tragically, it wasn't the disease that killed him but being shot along with the rest of his family after the Russian Revolution. WAH!
Robert Wyatt probably fits this trope, though to what degree (and more pertinently, what effect his paraplegia had on his music) is conjecture.
Spanish Opera singer Jose Carreras had to temporarily retire due to leukemia. He got better and came back to business.
Ringo Starr of The Beatles was very ill as a boy, according to The Other Wiki. He contracted appendicitis, at six which put him in a coma, and had a bout with pleurisy at thirteen, which led to him being placed in a sanitorium. These setbacks led to many missed days in school, and his education suffered from it. He also fell ill during the Beatles' Australian tour of 1964, and Jimmy Nicol took his place temporarily. Later on, he had many allergies and sensitivities to food, and brought his own food with him to India in 1968.
Ian Dury, of the Ian Dury and the Blockheads, had poliomyelitis in the age of seven and was paralyzed from his right leg. He was active on disabled people's organizations in UK. Sadly, he passed away from intestinal cancer in 2000.
Japanese singer Kouji Wada, who recently had to put his career on hold to go through cancer treatment. To make things worse, it's the second time it happened.
Noah Antwiler is currently this - in one of his video blogs from November 2010, he revealed that the reason his schedule has been slipping, and the reason why he looked so unapproachable during his E3 trip, was because he has a heart condition that was making him severely ill.
And now his depression has his sleep cycle out of whack among other side effects. The guy needs a hug.
Suffering from severe bronchial asthma as a boy, Alan Young was bedridden for long periods of time, up to four months per year maximum, thus encouraged his love of radio.
We "owe" the invention of the jacuzzi bath tub to one of this, actually. An Italian-American boy named Kenneth "Kenny" Jacuzzi was diagnosed with rheumatoid arthritis when he was around a year old, and from then on had to go through hydrotherapy treatments; his engineer father Candido tried to help him via making him a bath tub with whirlpool pumps. The rest is history.
Noel Fielding has Gilbert's Syndrome. He has a pretty good handle on it, but it's forced him to bow out of acting roles occasionally. He also had Hepatitis when he was younger and spent a year virtually bedridden on Julian Barratt's couch.
Nicholas "Noodle" Lee, son of voice actress Amanda Winn Lee, was born with infant leukemia. Her experiences with a child who spent a large part of his early years in a hospital are chronicled in her autobiography, The Noodle Chronicles. Thankfully, her son has been in remission for the last few years and appears to be well on his way to recovery.
The families of the aforementioned Saint Gemma Galgani and Saint Gabriel of Our Lady of Sorrows. In Gemma's case, her mother Aurelia and her beloved younger brother Gino (who was studying to become a priest) both died of tuberculosis before she kicked it in the same way, while the her eldest brother Carlo died as an infant. In Gabriel's, his mother Ines and several of his siblings (his sisters Rosa, Adela and Maria Louisa, and his brother Paolo) died of illness.