She sounds depressed and insecure. I would be inclined to enthusiastically accept the chance to babysit. Be careful not to criticise in any way. She needs to know she is loved and accepted even when she may not like herself very much, though of course do not suggest that last part to her.
This is not about you or your husband, even though it feels very much as if it is. The girl is unhappy and using whatever means she can to make other people feel unhappy too. She is unwell. This is hard to do but bite you tongues, be laid back and welcoming and let her know that you are always there for her. I would say that it might be unwise to just turn up without warning or invitation. At the very least make a phone call a couple of hours in advance so that she can be prepared and tidy up if she needs to. Good luck, hopefully time, patience and unconditional love will create the solution you want.