Honestly that situation sounds stressful and you need to focus on you and your baby. This is your time to shine and be happy. Even though the little girl isn't at fault it sounds like the mother has some issues she needs to resolve. At the end of the day it is up to you but think about what is best for you and your little one first.
If I were you, I would reach out to the little girl. You never know, but you could be the best thing that happened to her. Her experiences with you could change her life for the better. I had a friend from Russia who was a nanny to a little boy. This boy had had several nannies previously, and he had had terrible experiences with them. One ignored him totally and concentrated on Internet dating, the other also ignored him. So my friend made a huge difference in his life. She's been gone for several years but he still talks about her like she's still around. I think she still keep a contact with him.