Hmm, okay now you have a real problem here.
Teenager can be exhaustingly difficult, I know it, because I was one, you know it, you were one, too. In my youth I had very similar experiences to this girl. My parents quite forced me into learning the piano (with a very unfriendly, demanding, "it is all about technique" teacher), and I hated it. Luckily, that was in my pre-teens, so that I would not resist to much, but there were some conflicts, I can tell you. Then, I think I was 12, I got a new teacher, very frienldy, funny, nice to be with, you know. Still, I could not enjoy playing the piano, because I had the notion that piano was
forced on me by my parents. The result was that I would never do as much as I could in the lessons with her, would not prepare even the most easy pieces sufficiently (I admit it was not that bad as with your student, but quite similar). To that time I just knew that I would have to stop piano when there was a chance and I looked forward to that day.
Now, two main events occured that changed my attitude completely (that is probably the helpful part of my post

). First, this teacher was extremely patient, friendly understanding. She never forced me on anything, gave me all the choices I would have. I remember once playing Shostakovich and hating it, and after playing it like three months, she said: "You do not like it, do you?" "No." "Well, okay, then stop it, we play something different."
At the same time she was so sympathic, that I really
had to like hear (not because she wanted to be friends with me, that is most dangerous for your relationship with her, do not do that. Teenagers feel it, they hate it, they hate you. She would tell me some things about her relationship to the music, how she would feel the pieces, how she was impressed by her father playing cello making her cry, her experiences in college, etc.).
As you might presume, that really led nowhere in my development as a pianist (technique-wise), but it would make the lessons bearable for me.
Now, eventually I really liked her, but still was not very excited about the music. You cannot force love to the music. The girl may like you in the end, but she will never admit it when you force her to play the piano in any way she does not want. And she probably will not accept the beauty of piano playing until she has some distance to her current situation.
The second event was my teacher getting a baby (not from me, stupids). She had to stop teaching for two years, so I would not have to look her in the eyes when telling that stopped (that was pretty lame, but I was like 16 then

). Eight years I was forced to play the piano, and now I was free. I would still play the "Entertainer" some times, but that was pretty much it. My parents accepted that I would not continue to play, stopped urging me, you get it. Anyway, three months later I heard the Aria from the Goldberg Variations played by Glenn Gould in a movie. I loved it. Then I heard a pianist who would make a comedy show with all the most fanciest stuff with classical music. I heard Beethovens Fifth and started to take interest. I would hear Mozart on a TV programme. Some weeks later I had thirty CD's of classical music and a bunch of new notes. I started playing some real stuff (without a teacher) like Beethoven sonatas, could not do them, but wanted, and practiced all day. I made progress (through other pieces) like I never would before. One year later I could sightread through the pieces I had struggling with with my teacher.
Today I am thankful that my parents forced me to play the piano, that my first teacher taught me some technique. But I really love my second teacher for giving me so much advice and really giving me her notion of experiencing, living music, with words I still remember, I can understand today, but would think of as nonsense back then.
OK, now that was pretty a long story, so here is my actual advice (
note that this is just based on my personal experience and how you told your story, it may be very probable that all I am saying does not apply to your student).
1. Be your yourself. Completely. You seem like a frienldly, knowledgeable, caring person, and I can perfectly imagine you are a great teacher. Be friendly, laugh, try to make the lessons as nice as they can be for the girl. Do not try to make friends with her, do not try to make jokes all the time, or anything. (Really, that thing about "I like your hair" was most stupid. She must have despised you

)
2. Forget about progress as a pianist (concerning repertoire). All you can do for her is to get rid of this notion "My mom is forcing me to play the piano. It is bulls**t, and I hate everything involved in it". Give her cool pieces (Jazz or Blues is very good, if she likes it, also pop tunes. Best would be classical tunes she likes, of course - I really enjoyed Rondo alla turca back then)
3. Never, never, never try to force her. She may accept your order, she may do what you want, but she will remember it, she will hate you, she will hate the piano. Believe me, teens are like that. Do not give her the feeling that she is just a kid, don't underestimate her. Especially clever teens have this belief that they are the only clever ones in the entire world, all others (adults!!) being just boring caught-ups in everyday routine...
4. Accept the notion that it is probably impossible to stir up some enthusiasm in her, that piano is not the real thing for her - it may be the best for all people involved to stop teaching her

. On the other hand, perhaps her mother just will bring her to the next teacher, so if you feel that it is actually possible for you to lighten up the life of this girl, give it a try.
5. Speaking to her mother can be adviseable, if you want to get rid of the kid (trying to persuade the mother that piano playing is not the best for her daughter). If you want to try it with the girl, as I see it, talking to the mother will be seen as a "conspiration" against her and she will never accept you...
Okay, now that was really long, and I fear it is all useless stuff, but anyway... Now that I have written it, I will click on the post button...
Best luck with your student,
