If someone has Alzheimer's Disease but has led a pretty full life are they in Integrity or Despair?

I studying Erikson's theory and am trying to write a paper on a particular person but I can't ask her any questions because she is at end-stage Alzheimer's.

I do not think that Alzheimer's determines if the person is in integrity or despair. You mentioned that the person had a full life so I would guess they were (are) in integrity.

Wow, what an interesting question... To my knowledge, Erikson's theory doesn't really take such issues into account. It's just a conceptual framework into which all cases are supposed to fit neatly, but clearly do not.

It would seem that someone with Alzheimer's doesn't belong in either Integrity or Despair because they are no longer really "themselves" and can't make sound judgments about their emotional state.

If I were you, I'd pick another person to do your paper on and make your job easier.

Neither. People with Alzheimer's forget themselves, their spouse, family, friends, and many other things. They may even forget that they led a full life and at certain stages of Alzheimer's (especially the end stages), the person won't even remember that they're diagnosed with it.
My gramma has Alzheimer's and I am one of her older grandchildren who fortunately remembers how she was when she was in her 'normal' state of mind. I have seen the effects Alzheimer's has on the person who has it and their loved ones.
My advice is to pick someone who was close to her and ask them some of your questions. They may remember more about her than she does!