Simple answer: since the family did not provide a W-2, she files as self employed, reporting the income on Schedule C. You should be saving 35% of her income for taxes (15% self employment tax, 15% federal income tax and 5% state tax).
Essentially, the nanny and the family should agree, up front, on how taxes will be handled. If the nanny will file as self employed, she should expect to be paid about 8% more to compensate for having to pay both the employer and employee shares of FICA tax (social security and Medicare) tax. It is not unusual (but is not required) for the family to pay both the employer and employee shares of FICA, so asking for 15-16% more salary would not be unreasonable. See Opus 17's reply for details.
