I have not been in your situation so others I'm sure can offer more personal advice. Listening to you describe your daughter, however, reminds me of an abused woman and how when the abuser apologizes and is OK for awhile she continues to forgive and to go back to him. Is there a woman's shelter in your community? They have a great deal of experience with these things and could offer you guidance. Have you seen a counselor? Any chance your daughter would go to a counselor? I wish I could be of more help. Let us know how things are going for you.

I made an appointment for her with my therapist who specializes in adolescents but my husband balked...so the therapist suggested family therapy where he is included--so we have an appt.
I went to see a divorce attorney to see how I can legally get her to stay with me and he wasn't very encouraging...saying it would take suing for custody, and that the judges almost always let a 13-year old girl stay where she wants--if she's forced to go where she doesn't want to be, he says,many times a teenage girl will find an older boy...run away...get pregnant, etc. he said it's totally different than teenage boys. I'm not sure I agree with this--

It sounds to me like you are concerned about her safety. That is a completely different thing than just not wanting to give in to her. I still think it might be wise to seek advice from an abused women's center. Good luck. My heart aches for you.