I think it's fine. I agree on you providing the car seat and making sure it is properly installed. Did you do a background check on her? I was able to also do a driving record check on our nanny even though she probably won't drive our kids around until they are a lot older.
Just ask yourself what the difference is between letting someone drive your baby/toddler, vs. 10 yr old. It's no more dangerous, and it's got to happen at some point.