Go see an attorney. If there is no custody/visitation order in place (what? For 15 years neither of you ever pursued legitimizing this child?) you need to get one. Where you file for it has to be determined... you said "fly" home.... is she living with Dad in another state? You will likely have to file in THAT state for custody.
But at 15, she is going to get a say in where she wants to live, and I doubt the court would not follow her wishes... but there will also be future visitation ordered. And child support. So dear daddy is not going to be happy about any of it and will fight you every step of the way, I'm sure.
Talk to an attorney. You are going to need one.
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Just noticed you said she's been there 4 months. You might be able to act in your home state, if you act immediately. Typically, the child's residency determines which court/state has jurisdiction, and that residency is determined by where the child has lived for the last 6 months. When 6 months are up, you may have to go to the courts in his home state. Talk to a local attorney, but act now. Don't delay.
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After your SWH addition:
I don't follow, or rather, it makes no sense to me. He started paying child support when he started visiting her regularly when she was age 10-11? Why were the courts involved then if you just sent the money back, why did you ask for it? Were you a recipient of state benefits?
Changing schools mid-year is something you should discuss with her dad, and with her. At 15 she's what.. in 10th grade, right? Is she in between semesters in her current location? Or does she need to finish up another few weeks before she'll receive credit for this past semester? Teens don't always look at the bigger picture. Maybe her dad is doing that on her behalf, like a good father. What I don't understand is if you have such an easy going relationship with him for all these years, why do you think he suddenly has changed? Have you not discussed any of this with him directly? It sounds almost like you sent her there and haven't spoken to him at all, just your daughter. I don't get that. I mean, if you always had a contentious relationship in the past, I can see limiting contact to only what was necessary to effectuate visits and keep a relationship with his daughter viable, but if not contentious and you are trying to be good co-parents, then why do you not have more information/understanding about why he isn't sending her home when she wants to go home?