OP is upset. They thought they had a great nanny, but the nanny has decided to quit without giving notice. I can understand that they feel upset and betrayed.
OP, you could sue, but what would you gain out of it? If the nanny is in another country, good luck with ever getting money out of her. It is going to cost you much more than it is worth to sue her. Your time, money, and emotions would be better spent focusing this situation into looking for a new nanny. Unfortunately, employees, no matter what the job, are going to leave employers high and dry. It sucks, but what good is it going to do you to hang on to the feelings you have? You are entitled to your feelings, but don't get stuck there.