T.G.
What consequenses do the children have for misbehaving while under the nanny's care? (besides an appology letter) Do they loose priveledges like tv, computer etc?
I think you need to ask the nanny to sit and talk with you. Find out what is going on that warrants her to want to leave so soon. Then you need to address the issues that are bothering her.
Then you need to sit the children down and find out what is going on from their perspective. Do they really want her to go, and if so why?
some times there are just personalities that don't work well together. If there are no serious issues that warrant you letting her go, I would take the nanny's concerns and address them seriously and try to get her to stay a little bit longer to see if she is a good match.