*hug*
Well, let me start by saing that she is not a very nice person. There are some terms I would use to describe her that I try very hard not to use anymore, I got kids.
And don't get yourself down. It's been an eternity since I tutored someone and it was in every case easy to see when someone actually wanted to do better. I never got any grief over not helping a lot but one of my friends at the time actually expected me to do her homework (she was studying medicine at the time and did not like the chemistry aspect of it which was incidently what I was studying) and got storming mad at me when I refused. It's basicly the same thing. People expecting problems to magically disappear without any input from them. Especially if money is involved.
Anyway, tutoring is not an easy job. And I guess that few students ever appreciate that. You should really work on listening to your friend. You are not responsible for others failures, you can only do so much and if the other person acts as if you are responsible for walking the whole way, they will never reach their goal. It is not the nicest thing to say, but you should try to be more cynical or you'll inevitably end up with a burn out. Focus on the ones who were receptive for your help and not on those who don't really try. Do not take your work home, especially emotionally. Your students won't die from failing and if you can't seperate yourself from their failure you'll end up unable to help anyone. My husband is a nurse, similar effect there. And he can't stop complaining of his colleagues who work themselves up trying to help everybody, then burning out while giving the administration the feeling that they could run the hospital with even less nurses and paying them even less. Ok, that was a bit off, but the point is that you'll need to be able to still help yourself in order to be able to help others. And that is probably as hard as tutoring someone really stubborn.