ikno, you seem to be really caring and devoted to your students and frustrated that you are not connecting to this girl.
First off, I think said student really *needs* your approval but she’s unwilling to admit to this right now and she is willing to go to great lengths just to get your attention. I’m guessing that she somehow got used to getting negative attention all the time and this is her way.
See if you can find anything positive – anything, no matter how minor you think it is, to help encourage and praise her. Focus on that. Praise her when you catch her doing something good and tell her how amazing her good qualities are.(even if it’s something like remembering to do her homework) Eventually she will realize that it’s useless to disrupt your classes because she’s getting positive attention from you anyway.
If all else fails, and you’ve tried different techniques for another little while, then I would see this as a cry for help, in which case I would suggest you proceed with caution. When you do, ask your principal if there’s an expert on complicated chinuch matters on the school board, or if the school has someone to consult with on this matter.
I wish you good luck with this, and all other chinuch challenges that come your way as a teacher. I have tremendous respect for you as you seem to be very caring, compassionate and serious in your role as someone on the front lines of chinuch and literally working on behalf of Klal yisroel.
Hatzlacha.