No, she is not a household employee. She is self-employed as a caregiver/babysitter/nanny, etc., doing business as a sole proprietor. You do not need to issue a W-2 or a 1099-MISC Compensation form to her.
She has to give you her Social Security Number or Employer Identification Number (if she has one) and the dollar amount you paid her, so you can complete Form 2441, Child and Dependent Care Expenses - IRS.gov, on your personal income tax return.
To enter your daycare expenses, click on How do I get to the Child and Dependent Care Credit in TurboTax?
