Couple of things... she's paying you cash and I assume you are not claiming this as income. You also are NOT running a licensed daycare- meaning that you are not paying an annual fee for insurance and licensure. So... if this is all true, you would not be entitled to the same rate of pay as those who ARE claiming, insuring and licensing.
Having said that, I pay our date-night sitter $10 an hour. My son is 3.5 and completely potty-trained. She's a wonderful college student who drives very far out of her way to take excellent care of our child.
In your case, I would think that $5-$8 an hour would be reasonable. Add it up and see if that covers your expenses AND reflects the work you are doing with her child. Just hypothetically... if you are watching the girl for 30 hours per week, then you should be asking for $150-$250 dollars.
Personally, I would call around to local in-home daycares and ask what they charge for a part-time child. Per diem rate is usually slightly higher than a full-time kiddo. Get several quotes, type them out for mom (including the name of the daycare and the rate). Then factor-in the fact that you are working for her in cash (so figure out the taxes, licensure fees and insurance fees that the others would incorporate). Have this all ITEMIZED on a spreadsheet or something else that looks "professional".
Then... add in your "expenses"...
- gas mileage
- food
- diapers
- wipes
- extra paper towels, etc.
Show her the discrepancy between what she is paying you and what your final "loss" is. Allow her to look at the information in comparison to what she would be paying elsewhere. Remind her that very few places will take a child as early as 5:30 in the morning.
Then ask her what she feels is reasonable. Perhaps she increases your pay, but starts sending in food and other goods to offset your costs. Fine. Great. As long as you are making money in the end- that's fine.
Be willing to negotiate, but don't continue to watch this little girl for nothing- even worse, a loss!