Hi schnitzel,
I am really sorry for your current situation, first of all.
I'd like to summarize the situation:
-Your piano teacher is ill, and is unable to live without you.
-You don't enjoy the piano lessons, because she relies on you too much.
-If you stay with her, then you will feel bad for your mother, and if you leave, you will feel bad for your teacher.
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This is an incredibly difficult situation, and I can see your dilemma. I am not a psychologist, so I will not be able to give you the best opinions and advice... but from a 3rd person point of view, I think you should leave and find a new teacher.
The reason being is, your main reason of taking piano lessons is because you want to learn piano. If you are not learning from piano lessons, then what is the point of taking them? She will feel bad about losing a student, but if she cannot do the job, then in my opinion, she should not be doing it because she is not doing the job, however taking the same amount of money. About your piano lessons, I strongly feel that you should get a new teacher, who you can be confident about.
That doesn't mean that you should leave your teacher. You can still take care of her. Buy fruits, whatever, if you have the time and effort. You can recommend her to go to charity organizations, and receive outside help. Leaving your teacher doesn't have to be not seeing her forever. Maybe ask her family. There is absolutely no reason you should be her caretaker if you don't want to do it.
As for the dilemma about money; you feel bad either way. If you stay, you feel bad for your mother, if you leave, you will feel bad for the teacher. However, if you leave, the money will be used more productively. So that's why I think you should leave your teacher and find a new teacher.
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Again, my opinions are just opinions. Before doing anything, I suggest you listen to other people in this community as well. You can discuss this with your family too. I have never had a hard situation like this, and I hope you can come up with the best solution

Always stay positive!
Natsu