Hey Tizz,
I recognize many of your daughters traits with those of my own. I hate to do anything that's not interesting to me.
But here's some of the difference. I obviously matured too quickly. So it would probably help to treat her like an adult, also by her teacher.
I don't remember what actually motivated me at that age, I think I was quite average. Actually, I think I spoke a lot in class, and I most definately hated homework.
Around the age of 13-15 I had grades, and I was driven by the consept that I had to do well to become rich as an adult(so I could provide for my family). I did not understand the meaning of the things I learned! And this is where my teacher was great, because he respected that. But he said I needed to get through with it, I had to learn it, so I could get into a good school and become successful. I'm not saying that your daughter is at this stage, I'm saying that some of it might apply to her as well.
Key is to have a good teacher-pupil relation. Even though she's gifted, she still has the same rules as everybody else. Forget about being gifted, think of her as a normal child with strengths and weaknesses. And ask yourself how to prevent those weaknesses from taking over. Think motivation and encouragement.
Below is something that might be a bit off, but might be relevant.
I don't know how you're raising her, if you're giving her a lot of space to do whatever. The thing is consequences. I was brought up getting a lot of slams on me. It might have been very harsh. I don't have anything to compare to. My parents were strict. So I think that has made me into the person I am today. I am thankful for what they did to me, because without it I could have been an idiot. But consequences are good sources of motivation, being in contact with the real world.
Give her the truths. Tell her, that she has the potential in her, if she wants. Tell her how much you love her and how you'll always have her back.
You say that she says she is the smartest, and I believe she is, but maybe she has gotten ahead of herself. She says she knows it all, well, present her problems she can't handle yet. Don't give her worksheets, give her a math puzzle that will make her rethink her position. I know that math problems that I can't solve, will drive me to learn how to solve it.
Anyway, this is just my perspective on things

There are many thoughful and great posts here already. A bit hard to get a grasp on things, but I hope I helped just a little bit.