R., my heart goes out to you.
My situation was somewhat different, but had a similar underlying cause. My younger daughter (now 22 and thankfully well-adjusted - who knew she would ever be ok?) saw her father about every five or six weeks, even though he lived a couple blocks away. He just didn't seem to be able to make time for her, he didn't accept her artistic nature, his new wife didn't like her, etc. She felt SO REJECTED by her father, and she had terrible self-esteem from that. She also cut herself, became bulemic, was angry, unhappy, difficult to deal with, etc, and it wasn't really because she was unhappy living with me (single parent at the time). It was because she felt so rejected. I think this sort of problem has to be WORSE when it's the birth mom who's rejecting her. A girl relates to her mother, and probably doesn't get why her mother wants so little to do with her. She is venting it on the only consistently safe people she can - you. You and your husband have been her stability, and so that's where it's safe to take out all that anger, etc.
I suppose it doesn't have to be said that you need an excellent therapist for her. If at all possible, the therapist should get her mom involved in the therapy. If the mom is asked to visit the therapist to tell the therapist her observations around the daughter's behavior, then the therapist can take it from there. You don't need to tell the mom, "you're screwing up your daughter, so you need therapy too."
You might consider getting a therapist too. I spent a lot of time with the therapist asking "am I parenting well?" "is there anything else I should be doing?" "is this my fault?" etc. Watching your daughter fall apart makes you feel helpless and like a bad parent. I needed the emotional support too.
Anyway, I'm sure you'll get some very good ideas here. Good luck to you, and do let us know how things go. I still remember those days and 13 was just the beginning.
You're in my prayers...