Would you take her back?

From a woman's point of view, I would NOT take her back. It sounds like she is running to you because she doesn't know what else to do and feels like you are a "sure thing" as far as giving her and her kids somewhere to stay while she figures out what her next move is going to be. I don't want to tell you for sure "don't do it" because I'm sure there are particulars that I don't know, but it sounds fishy to me. I'd stay away if I were you.