I have some experience with kids like her as I once worked in a kids day treatment program as a teachers assistant. I don't know how much I can help but I'll try. First and foremost, has this girl ever been seen by a psychiatrist? My first impression is that she is either very manipulative or may have something like oppositional defiant disorder. I am not a doctor or a teacher, so these are mere speculations on my part and I haven't actually been around her, but I think I can offer a few suggestions.

The first question, I really don't know how to answer except find creative ways to get her to try things that will give you insights to her strengths and weaknesses. I'm thinking along the lines of board games like scrabble, or chess or checkers. Uno or mancala are pretty good too. Such activities can tell you a lot about how she thinks, if she's honest, her level of patience, etc. All the things you need to learn about her in order to determine the areas she needs to work on, then work on those things so that you have a better idea of how to help her learn. It will also help you understand if she is a visual learner, which I'm sure you know makes a difference in how you approach her.

I also think before you can even begin to teach her anything, you need to let her know in no uncertain terms that if she wants the reward, she has to do the work first. This is NOT easy, especially since you basically got off to a bad start, but it's still not too late to STAND YOUR GROUND. This is the hardest part. When I babysit my grandson, he thinks because his Gramz likes to play, that naptime is playtime, too. I have been know to tuck him into bed, which lasts for a few minutes, then he gets up and I put him back, then he gets up again, and I put him back. The first few times this went on for probably a half hour or so. Then he tried crying, then screaming and telling me he hated me. I ignored all of it and eventually he wore himself out, realized that Gramz was no pushover (well, except about ice cream, I always buy him ice cream, LOL) and he might as well just go to sleep. It took extreme patience on my part, and I have used similar methods with students and it has always worked - eventually.

The fact that this girl thinks she has power over you is the one thing you have to fix before you can proceed with anything else.

This fear of heights thing? You mean she refuses to go upstairs at the library because the kids books are on the second floor, if I understand correctly. That's simple, if she refuses to go upstairs, just sit down somewhere near the top where you can still see her, and wait. Again, the first time you may have to wait a while, but eventually she'll get the message. You have to take all the fun out of her refusals to cooperate by basically ignoring it and act as though you could care less how long it takes her to make up her mind to cooperate.

I have to say, I don't envy you. I've had chairs whipped at my head, called every name in the book, bitten, and stabbed. The hardest thing is not reacting. Sounds crazy I know, but it works. Kids like her don't know how to respond when adults don't yell and get mad at them. If a kid throws a chair at me, I just calmly tell them to pick it up and put it back where it was. It teaches them to trust you, that no matter how badly they behave, you won't walk away. Obviously she knows that adults will always walk away, because all her previous tutors have.

Hope this helps.