MB here, so I might not be as helpful as a nanny, but how about:
- ask their history in hiring/retaining a nanny
- how they would describe their parenting style
- ask what is most important to them in a nanny - what are their "must haves"
- ask about the personalities of their child(ren) - what are they like, what do they enjoy, etc...
- maybe ask what a "typical" day in their house would be like
Obviously all the basics of compensation, benefits, hours, etc... But hopefully they'll be good about being organized and professional with their side of things.
Good luck!