The legal way to do it is to compensate her (at the IRS rate) for actual mileage.

You could very easily just track the mileage for the roundtrip you know she'll make 3x/week and therefore know the monthly amount for that, and then have her track anything additional.

We just started with a new nanny and none of our other nanny's have ever needed to drive anywhere. We live in Arlington and there are plenty of activities within walking distance. However, my oldest is in preschool 3 days a week. The school is about 1.5 miles away and she will be taking the kids back and forth. Also, once in awhile I am sure she will drive a mile or so to the library. I would like to just give her a flat monthly reimbursement for gas rather than adding up each mile and deciding on a rate. How much would you compensate a nanny for this type of minimal driving?