Hun, if the other mother wants to take her on, let her. But SDD should pay her own board, I have no idea why you would pay it for her. She can't do things half assed and either can you.
She should sort the house rules with the mother, and what SHE says goes, not you. It's HER house, not yours and I think the friends mum has a right to decide what the rules are.

She isn't going to prove she can act like an adult until she is forced to.

Don't forget she has treated you all terribly, she is old enough to do that therefore she is old enough to learn not everyone will put up with it. Also, sheis smart enough and ballsy enough to lie, cheat find a place to go on the weekends - she will be fine fending for herself xoxoxo
I agree with everything Lulu said.