I'm unclear as to what you need advice about. Is it the fact that your daughter is living with you, yet would rather live with her abusive mother, and that you don't know how to get her to enjoy living with you?
How old is your daughter.
Why do you keep referring to her as "the child", instead of "my child" or "my daughter"? You seem to speak about her as though she is a stranger!
I noticed aswell that you said "I think the world of her" but not "I love her". DO you love her, or do you just like her a lot?
Could it be that maybe the reason she wants to go back, is because although her mother is awful to her, she is a familiar person. Someone she has known from birth. Whereas I presume that you maybe haven't been around much in her life? (forgive me if that's incorrect). Maybe she's desperate for some stability, and the only thing she's used to is her mother, despite the abuse she gets from her.
I'm no psychologist, but as a mother I do know that kids thrive on routine, and being with you now is obviously a world away from anything she experienced before with her mam.