First of all, POOR KID! although it is NOT your job to save the child, how could anyone not feel for her? Her situation is not one that any child should have to deal with. How long did you say she has been with you? Do you have enough authority over her that she will listen and respect your correction? or is she a loose canon? this child needs to be told that this is not the way we treat ANYONE! it is not acceptable for her to disrespect any adult. Or any other person. She needs to be shown how to love, she is obviously not getting the proper training at home. sounds like she is coming from an abusive environment! If you cant keep her under control though, I do think it would be grounds for termination, however....the problem is NOT the childs behavior, the problem is how and who is raising the child. If you cant keep a grip on her while she is in your care God knows you will never be able to stop what goes on at home, looks like a losing battle on your part.
As far as the Mother coming to you about personal matters at home? I would inform her immediately, you are no longer a confidant or "go-between" in their marriage. You are there for one thing and one thing only, and that is to take proper care of their child while they work. This has happened to me before a few times, I was in the middle of a couple of messy divorces. Although we sometimes tend to form a bond with the people we work with, the bottom line is it is strictly professional when it comes to some things, and marrital issues are OFF THE TABLE!!! Good luck with whatever you decide. Dont forget though if you are going to let them go, dont make it directly about the childs behavior alone, they already dragged you in so you may as well let them know it about the childs behavior resulting from what seems to be going on at home. That would be MY one last ditch effort to nake them realize what they are "doing" to their daughter. SO SAD!