After reading all this, I am stunned that wife, who works with troubled kids, hadn't insisted (long ago) that this child have a neuropsychologist evaluation and receive help much earlier than now. This (teen years) is a bad time to try to start it. I'm betting she's involved in drug use. But even if she's not, she's running the show and is also a very unhappy young woman. I had a very unhappy teenage daughter too and she ran away too, but we never ever were lenient about it. The cops got called...she was brought home. We put bars on her bedroom window so she couldn't leave out her bedroom window and run around at night (yes, she did this). You may need to put an alarm on her door...they sell them at Radio Shack. It is DANGEROUS for her to be running around at night. My daughter is now 26 and those days are long behind us, but it wasn't fun! As soon as she started acting out, the money dried up. We didn't give her any.We bought her what she needed, and that's all. No toys, computer, phone, etc. She got a job at sixteen, which helped her current excellent work ethic, because she wanted nicer clothes than we were willing to buy her. She also wanted junk food...we only bought what she absolutely had to have. She pouted about that a lot, but oh well.
wife went through a divorce so SD has another parent and if he is not on board, daughter may actually end up living with him. A lot of time kids, who are very disturbed by divorce, often play one parent against the other and he is allowing it. In certain states kids can decide who they live with at 13 unless the parent is proven unfit. wife needs to go to therapy herself so that she can learn to be a whole and happy person even while her daughter is going through this and trying to make her unhappy. EVEN IF HER DAUGHTER ENDS UP WITH HER EX, she needs to realize she has a life too, aside from her daughter. It's unlikely estrangement will last forever.
In the end, my guess is SD will live with her father because she has apparently inherited and/or copied his personality and he sees nothing wrong with it. But wife should have had this child evaluated years and years ago...if she raged, broke things, chronically lied, etc...she knew that wasn't normal and must have been in denial. The sad thing is, this issues only get worse as the kids grew older, especially if there is chaos in their lives.
I found most psychologists to be unrealistically lenient regarding teen age behavior. One therapist told me and hub we needed to "trust" our daughter. Huh???? She could look us straight in the eyes (and did) and lie with tears. She did drugs (daughter has told me since getting clean "never ever trust a word a drug user says."). Is SD's father a substance abuser? Do you think SD could be using drugs/drinking? They usually do it when you are asleep. If so, and even if not, I'd be extremely tough on her, even if she still rebels. My daughter tells me all the time that we did the right thing when we got very tough on her and that she used to "play" the therapists for sympathy.
I wish you all luck, whatever you decide to do.