When I was your daughters age I went through a similar phase with my Mum (minus the stealing). The difference was it was a new baby in the house not a new boyfriend but the result was a bit similar. Understandably myself and my younger sibling were put out by the new baby but my mother did nothing to ease our jealousy and instead reacted with hostility. She'd say things like if the house was on fire she'd save the baby and we could fend for ourselves. Now I'm adult I understand more what she was saying but it was the way she said it and when you're 13 you don't always think rationaly!
Anyway our relationship deteriorated more to the point where I just didn't like her anymore and I withdrew and stopped looking for her attention. So she started looking for mine, in a bad way. By the time I was 15 we were fighting every day, we both felt rejected I think. I was moody and sulky alright but at the same time she would pick a fight with me like if I wore my hair a different way she'd say something like "oooo look at the state of yer woman today" in a bitchy voice. Or just pull a face at me and when I'd ask her what she was making a face for she'd start getting hysterical saying she hadn't done anything. Or mumble something and then go "oh nothing nothing" and smirk. Then she'd go squealing to my dad about how difficult and moody I was.
The reason I'm rambling on there is when you say your daughter is treating her mother badly and not doing as she's told have you witnessed this you Self or are you just being told by the mother?
My father only heard my mothers side of the story. Once he threatened a talk about "this attitude towards" her but it never happened. I was ready to tell all even if I got a bollicking.
I was always a Daddys girl and my mum resented this as well. Chances are your ex is aware her children now prefer your company to hers and could be trying to slander (for want of a better word) your daughter in your eyes. If I were you I would sit down with your daughter and calmly talk to her about the problems. Give her a forum to air her displeasure in a safe enivornment. Say something like "for the next hour you can tell me anything and I won't tell your mother and I won't mention it again". If she starts saying things like "mums a slut or a bitch" don't give out but just calmly say "I know you're angry but you really shouldn't say that about her". You might get an insight into her world and advise her on how best to deal with her anger. If she is being unreasonable herself then she might see that for herself after talking about it. You might get met with a total brick wall either but it might at least help open the dialogue further down the line. If you can't do this, is there an aunt or family friend she could talk with?
It's not fair of your ex to push this back on you but at the end of the day theyre still your children and if somethings upsetting them then you need to be able to help them deal with it. Good luck!
Oh and PS I have a great relationship now with my own mother once I learned to see the bigger picture but I might have got there faster if someone had stepped in and helped me see it