There is a specific federal law that addresses the maximum deduction employers can take weekly for room and board. I will try to find that info, but I believe the max was around $130/week.
So no, you cannot require a nanny actual rent to live with you for your convenience. You might pay $2/hour less than you would pay her as a live out, but that is as far as you can go.
So $14/hour for 50 hours/week would be $700 gross unless you live somewhere that requires you to pay a LI overtime. If you include your employer expenses, a LI Nanny may cost around 40K a year. Of course, if the applicants are unsatisfactory when you offer $14, you'll have to up the wages.
An AP costs about 28K/year, but you might have a hard time finding one who is infant qualified and willing to care for 3 kids.