Mademoiselle was an idiot.
She may have tortured countless girls to find out about the afterlife, but she completely missed the point of martyrdom: to suffer and die selflessly for your beliefs. Jesus Christ and the Christian martyrs endured huge amounts of pain not because of innate tolerance, but because they denied themselves and were ready to sacrifice themselves for the philosophy or ideology that they profess, even if it meant suffering for it. Their belief was what gave them the endurance to withstand torture in the first place. Now, the film does reference the Greek etymology of the word "martyr" which means "witness" however the reason why Christian theologians used the word was in the context of the severe persecutions committed by Ancient Rome
: Christians were witness to the message of Jesus, but were tortured and killed for it, hence why the word martyr became associated with the "die for your beliefs" definition of martyrdom. See also: Inspirational Martyr
and Doomed Moral Victor
This was why Mademoiselle failed. The random girls she tortured, including Lucie, only ended up becoming victims and insane... because the girls Mademoiselle kidnapped were never selfless and had no other inclination or reason to make themselves suffer for it. They were never strong-willed philosophers or doomed moral victors, only ordinary people trying to survive and avoid pain. And such ordinary persons do not have the psychological composure capable of enduring trauma for prolonged amounts of time. However, Anna was highly selfless, and it was her selflessness and inclination to help others in need, including being the only friend of the otherwise-suicidal Lucie, that made her a Martyr. Even Mademoiselle disproved herself when the first example she showed was a Chinese man being tortured for being a political dissident.
Mademoiselle killed herself because Anna told her it was Paradise.
Knowing about the Other World without becoming a martyr bars you from ever entering.
Anna told Mademoiselle what lies beyond in the Other World which was likely a blissful paradise filled with everything a person could ever want. However, Anna tells Mademoiselle that knowing about the nature of the Other World without becoming a martyr bars you from ever entering, possibly being sent to Hell or oblivion as a result. Now that Mademoiselle knows, she can never enter, damning herself with no possibility for redemption. Realizing that she can't tell her followers because they too will be damned if she does and knows she'll most likely be tortured until she does tell them, she decides to kill herself to spare herself this fate and to keep her followers from being damned. This is why she tells Etienne to "keep doubting" because if he knows, he'll never be able to enter the Other World. This is also why Anna slightly smiles at the end, knowing not only that she will soon enter paradise but that she got the ultimate revenge on Mademoiselle, denying her access to Heaven.
Flayed!Anna was just bluffing, and there is after all nothing beyond the Other World.
Whatever she said after being flayed alive was nothing more than a big lie to Mademoiselle. The reason why she endured torture? Because she knew that Mademoiselle will use her as the telephone to the afterlife. Considering that she was Lucie's best friend, and pitied her miserable condition, Anna may have hand a grudge against those who caused Lucie to be insane as she was. She knew that Mademoiselle was the founder of the philosophy movement but she also knew that the reason why Mademoiselle sticks around torturing girls was because she was desperate to find out about an afterlife. Thus, Anna told her that the afterlife is so horrible, so nihilistic, that there is no paradise beyond the material world, forcing Mademoiselle to go over the Despair Event Horizon
and end the cycle of torture once and for all.
Thus why Mademoiselle killed herself
. Anna has achieved so much apotheosis... to the point that she contacted this thing
outside the material plane and this was where she received information about what the Other World really was. Mademoiselle thought the Other World was some kind of Eden where human souls were capable of surviving intact, only to find out that not only did any Human Soul survive, they were completely subsumed by the Eldritch Abomination
who was the true nature of the Other World. The reason why Lucie and the rest went insane? Because they were not fit to see the True Form. Eventually the Abomination will manifest itself into the suffering of Humanity and destroy the world. Mademoiselle, having tortured countless innocents in her study of the Other World, realized all her crimes were for nothing, and hence why she shot herself.
They harvest energy from tortured young girls. Over time the girls become so delusional that they developed monstrous alter egos (Lucie's undead monster vs. Witches) Anna was perhaps a proto-Madoka, sacrificing herself to end Mademoiselle's cycle of torture.
- Mademoiselle killed herself because Anna has seen the future where her kind will be destroyed by Homura.
Martyrs are quite literally witnesses
In the credits there's a note stating that "martyr" comes from the word "witness", knowing that and the fact that martyr are becoming rarer and rarer, a guess could be that instead of being a direct line to the afterlife, martyrs are actually the afterlife's direct line to heaven and are people designed to "record" things that happen on Earth. Due to their strong moral compass/firm beliefs the martyrs are able to objectively judge people to see if they are fit to enter the afterlife (likely some kind of Eden). The fact that Mademoiselle probably destroyed her only chance to enter afterlife, she killed herself for ruining her only purpose in life.
Martyrs are capable of barring one person from entering the Afterlife
Or at least, this is what Anna tells Mademoiselle. This is why Mademoiselle kills herself and reveals nothing. Anna forces her to do this or get barred.