I wouldn't know why she won 't do it, maybe she wants to make sure you behave while her son is in nik, maybe they have a deal to watch you...
Social services have a huge list of qualified nannys in your area aswell as baby sitters. You can get a full range of references from families they currently work for. I think you need to show yourself and her that you can do this on your own.
Go through Social services for the sitter or accept that she won't do it and stay in for the night.
You could ask her why she won't but if she has hidden reasons you wont find out if she doesn't want to tell you.
Sometimes when people won't behave how we want or need them too then it's time we changed how we behave to make ourselves feel better. You may not like the idea of leaving your child with a stranger, but thats exactly what you will be doing when you take them to school and nursery....
Good luck.