Good for the OP. On top of the mother's entitled attitude, why would she want to leave her child overnight on a regular basis with someone she barely knows? That's strange, IMO.
Also, I know the OP has already taken care of the problem, but there are real financial considerations to babysitting in one's home.
One is taxes. The mother (if paying the sitter) may want to claim her child care expenses on her taxes. This means that she will report the sitter's SSN to the IRS, which will make the sitter have to declare her earnings and pay self-employment tax, which can be close to 50%.
The other is insurance. If babysitting in one's home, be sure to check your homeowner's policy. Babysitting could be considered a business by some homeowner's policies, and so if the sitter got sued by a parent, the policy may not pay. Additional insurance could be needed.
It's really not as simple as the mom dropping the kid off and tolerating watching someone else's kid.