Family members do give gifts (of course), and $6,000 in one year (if there were no other large gifts) wouldn’t be subject to the “gift tax” (or even reportable, as this article says). However, if the payment was actually made as compensation for the child care services you were providing, the government would call it taxable income.
Importantly, they distinguish between care which takes place in the provider’s (your) home, and care which takes place in the youngster’s home. You’re considered the provider of “Child and Dependent Care” in the first case, but a “Household Employee” (the "nanny tax" in your title—more info here, and here) in the latter.
For the return preparation, your daughter would need to file a Schedule H with her return (saying “yes” to the TurboTax question about having an employee in the home), report and pay the FICA and unemployment tax on that form (there’s an exception for payments to parents, but it doesn’t apply if she’s divorced and not remarried), and then issue a W-2 to you. (There is a link in our program to apply for an “Employer ID number,” which would also be needed.) You would then include that W-2 income when preparing your own return, entering it like you would enter any W-2, and that would be the extent of your obligation.
The links above have details on that process, and here's a more complete (official IRS) guide to the rules. I know you were probably hoping for a simpler response, #LIZZIEO, but I hope this helps!
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