If you want to remain friends with that lady, I would think the best course of action for you would be to keep right out of the child-raising project. Get to know the little girl on a friendly basis. Take an interest in whatever interests her. If you intend to become a permanent member of her family, treat her as a friend, not a problem. Don't ever try to discipline her, or scold her, or catch her in a lie.
Kids tell lies.
And accept her. Don't intervene when her mother is involved. Be a friend to them both and don't try to solve their difficulties with eath other. When you are with the mother, alone, and if she really does want the conversation to be about her problems raising the child, just listen, don't preach. And don't even let it develop into a big thing - mothers always have to vent to someone about their children.