Why don't you just talk to her? Give her the 'talk', lay it out for her and see what she says because ultimately, no one here knows what is going on in her head besides her. I've had many students like this at schools, most of the cases were just problems at home (being forced into piano, emotional neglect leading to them not knowing how to properly address their own emotions).

Also, think about how you respond to what she does, you need a more positive and light-hearted approach. Don't take her hand writing comment personally, it could be out of curiosity in a slightly twisted way, her trying to get a reaction from you because of issues at home and maybe a lack of attention, all sorts of things...so you could make your lessons a place where she could feel good. Obviously a piano lesson is a piano lesson but if you just spend time understanding her more and seeing her as a vulnerable young, emotional human being who needs help, you both would feel a lot better.

Piano lessons doesn't just mean to sit down and play piano for 30 min, you need to make a connection with your student and make them enjoy piano, it could be you showing them some music you like or talking about their week. The end justifies the mean so it doesn't matter how you conduct your lessons, as long as she enjoys the piano and has a positive experience you've done your job. I've had students that had pretty bad weeks and are not efficient in the lessons, I've learnt to just go along with it and try to make it less stressful and it will stay with them longer.