Frankly, I would guess she is benefiting greatly from your willingness to pay off the books. I would tell her you will pay mileage per IRS for 200 miles a month max, but that you are going to change her pay back to being fully on the books. But I am mean, and hate hearing about tax dodging nannies.
If you don't want to do that, then I would work out the total amount she is getting from you now vs. fully taxed and 50 paid miles per week and present her with a choice:
Nanny, we have run some numbers, and are willing to let you choose what you prefer regarding your off-the-books pay and mileage money.
We can either continue to pay you $XXX per week gross with 25% untaxed and give you $10/week for gas, which would be $ZZZ/week total take home, or we can put your wages back on the books 100% and pay you IRS mileage rate for no more than 50 miles/week. That would give you $AAA weekly take home money.
What method would you prefer?
And I would also suggest you discuss a "petty cash" fund, limited to what you feel is reasonable for activities and meals out per month, and institute a receipt based reporting method.
IOW, if she takes out $30 for X, she returns the change and the receipts in order to get her petty cash replenished.
And FWIW, I have taken my NK out to eat less than 10 times in 4 years. We pack our lunches whenever possible.
Good luck!