If you are age 18 or older and earn $1,700 or more from the same employer then you are considered a household worker and that employer is required to withhold income and social security/medicare taxes and give you a W-2.
If you make less than $1,700 from one employer or have income from various employers and your gross income from babysitting is $400 or more then you should report your earnings as a sole proprietor on Schedule C. You will also be required to file Schedule SE for the payment of self-employment social security taxes.
http://www.ssa.gov/pubs/10021.html
http://www.irs.gov/pub/irs-pdf/p926.pdf
http://www.irs.gov/businesses/small/article/0,,id=115041,00.html
If she does not cooperate and the IRS becomes aware of her failure to file and withhold, then she can be penalized and made to pay the back taxes plus interest.
If you want to file as a self-employed sole proprietor due to her refusal, then you would file Schedules C and SE with your tax return. If you will owe $1,000 or more when you file your taxes, then you may want to pay quarterly estimated taxes in order to avoid a penalty for underpayment of taxes during the year.
http://www.irs.gov/pub/irs-pdf/f1040es.pdf
Enrolled Agent
EA, QPA, CHFC, CEBS, CLU - 29 years experience providing financial advice