If you were a parent with 5 caregivers, would you be okay with this?
Just tell her you don't want to work more than 15 hours a week.
I'm confused, are you working 60 hours a week or 60 hours a month? And yes, you can require consistent scheduling from your employer, as well as a cap on hours worked. However, your employer can also decide to give preferential hours to your coworkers who are more flexible with their time, which could result in your hours being cut. If the caregiving position is your primary job, you have to expect that this may happen and either try to pick up more babysitting hours to make up for it, or go back to your agency to try to find a position that works better for your schedule.
As a parent I accept the fact that babysitters have more responsibilities (let alone a life) than being at my beck and call, and that they decide their availability, not me. However if I have a choice of babysitters to schedule and some are consistently more available than others, you can't blame me for scheduling the best available ones first. So if you reduce your availability, there is a risk you will end up being scheduled for less than your preferred 60 hours a month. I don't understand why you feel the need to explain you're 'not enjoying yourself' on days you can't work. You don't have to justify yourself. It's none of the parent's business what you do with the days you're unavailable. Last time our babysitter was unavailable, she had plans to go to a party. Guess what I told her? To enjoy herself!