It depends.
Before the patient became brain damaged, did they express any desire, verbal or written, about what they would have wanted? If so, don't we have an obligation to carry out that wish, given their condition as you described it?
If they did not express a desire, either written or oral, then we are confronted with a euthanasia decision. Which is a completely different topic for another day and another thread.
Likewise, when she said she wouldn't want to live that way did she, at the same time, agree to die the way she did?
Just a few questions that I really haven't seen well explored.
So if someone has a written directive that states, "Please shoot me in the head if I become incapacitated and can no longer take care of myself" then it becomes our obligation to fulfill their wishes? People do not say "Please, starve and dehydrate me to death for 13 days if I become mentally incapacitated." And even if they do so, is that the right thing to do? Kill someone just because they asked you to? I'm sorry, I just don't think that's right.