Good evening Erin,
I'm Doug, and I'm sorry to hear of the confusion. My goal is to provide you with excellent service today. I have been a licensed attorney for more than 30 years, and I handle social security law issues like yours.
Getting a quarter credit is not based on the number of hours that you work, or what your wage is. In 2016, you must earn $1,260 in covered earnings to get one Social Security or Medicare work credit. So if you work in 2016 for your daughter and are paid at least $1,260 in wages from which social security taxes are withheld, then you will gain the last quarter of credit that you need to qualify for social security benefits as well as Medicare.
If you work as an independent contractor (which means that your daughter will not have to withhold taxes from your income)---a self-employed babysitter---so long as you claim the income on your tax return and pay self-employment taxes---that is all you will need to qualify for the last quarter of credit that you need to qualify for social security benefits.
You may reply back to me using the Reply link and I will be happy to continue to assist you until I am able to address your concerns, to your satisfaction.
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I wish you and yours the best in 2016,