Drusilla is frail and weak when she first appears on the series. She was believed to have been killed by a mob in Prague, but apparently was only severely injured (How that mistake happens with a creature that turns to dust when killed is anyone's guess), to the point that normal vampiric healing couldn't restore her. Spike manages to restore her to full health by draining power off of Angel.
Darla was a high class prostitute suffering from syphilys and waiting for her death. Then the Master gave her a visit....
Charmed: The episode "Awakened" revolves around Prue and Phoebe discovering that Piper is terminally ill with the disease Oroya Fever. While real life symptoms of Oroya Fever include being covered in warts and rashes, Piper only experiences coughing, exhaustion, and fainting, all done adorably. Being the main character, Piper survives, but not until after the crying and good-byes.
Many, MANY of these show up in Soap Operas, specially Latin-American Telenovelas:
The female lead from the Chilean telenovela Corazón de María is an upper class Ill Girl with a severe Heart Trauma. She gets a heart transplant coming from a middle-to-low class bride killed in a tragic car accident right after her wedding. The drama starts when she starts searching for the donor María's family to thank them for the heart donation, and then she meets and falls for her handsome and hard-working husband Miguel...
Shannon (Maggie Grace), who is asthmatic and needs to have an inhalator handy. Sawyer once stole her medicine and tried to use it to bargain.
Juliet's sister is an Ill Girl whose cancer and resulting infertility is what drove Juliet to do some rather... illegal things as a fertility doctor. When Juliet goes to the island, she is afraid to leave her sister alone, but, with typical ill girl sweetness, her sister convinces her to go.
Merlin: Happens frequently with Morgana, most notably in A Remedy To Cure All Ills in which the Monster of the Week deliberately makes her sick in order to heal her and so win King Uther's trust.
The X-Files: Scully takes on this role for one season after it is revealed that the tests preformed on her during her abduction have left her with (probably terminal) cancer. In accordance with the conventions of the trope, her appearance is relatively unaffected by the illness, and the only visible symptom is a Deadly Nosebleed.
In How I Met Your Mother, The Mother becomes one during the course of the finale, and we see her in a hospital bed for a few seconds. In the end, she dies of her (unspecified) illness, leaving Ted to go back to Robin 25 years after meeting her.
One of the "Lost Loves" cases from Unsolved Mysteries dealed with two little girls in a cancer ward, one with throat cancer and another with ovarian cancer. The two became fast friends, and the first one promised that she'd make the other girl the godmother of her firstborn to help her deal with how she wouldn't be able to have children. 20-something years later, the first girl was still searching for her friend... And a later episode revealed that she eventually found her.
Hawking: Portrays the twenty one year old Stephen Hawking as more of a very smart Ill Boy than a Genius Cripple, unlike conventional depictions of Hawking as an older man.
Breaking Bad 's Walter White has lung cancer which slowly kills him throughout the series. In order to support his family and pay for medicine, he goes into the drug trade, using his knowledge as a chemistry teacher to make crystal meth.
Kamen Rider Fourze: Kengo suffers from an unspecified weakness that rendered him unable to use the Fourze system as well as the Power Dizer; he's actually been nicknamed "King of the Infirmary" due to the amount of time he's spent there (even when he's not just using it as an excuse to help fight the Monster of the Week). The Aquarius Zodiarts uses her healing powers to cure him about 2/3 of the way through the series. This is later jossed in episode 45, when his illness/weakness comes back full force.
Kamen Rider Ryuki: Kitaoka Shuichi, even though he's quite a bit older than the usual victim of this trope.
Mirai Sentai Timeranger: Ayase / Time Blue suffered from the incurable Osiris Syndrome throughout the entire series. However, after time has been altered near the finale, a cure for his disease is found.
Danny's asthma causes some characters to perceive him as weak - to this day Charlie is constantly worrying about him and trying to keep her promise to watch over him. Justified given that this is a lawless, unforgiving Crapsack World and other than plant-based medicines there's not a lot that can restore his breathing if he has an attack. In "No Quarter", Danny also uses this to his advantage, to get the drop on the mook Private Richards who's been harassing him.
"The Longest Day" has Aaron Pittman and Rachel Matheson meet Philip Blackmore, who has been dying of injuries ever since he fell off a horse. Rachel decides to lead the Blackmores into believing that they can save their son, and then just take off with Aaron and let Philip die.
In true Ill Girl fashion, it is also implied that there's no real possibility for River to ever be cured of her insanity; the best that can be done is short-term chemical treatments, as the damage to her brain is extensive and her body eventually rejects any drugs used to help her. Even after the events of The Movie, its not made clear whether she's fully healed or has only partially recovered.