A.C.
It doesn't matter if your daughter will be sleeping or not. Yes, if she does, it is a much easier situation for the sitter...but you still have to have a sitter. Think about any time that you had to change your schedule around an activity because it was naptime or the baby went off their "normal" schedule.
The reason it is $4 at the gym is 1) part of member perks and 2) it a group setting, not one on one. In California, the Federal Minimum is increased in some counties. I am in Texas, and it is easily $8-10 an hour for an adult babysitter (again in a one on one situation). I'd easily pay $15 so that I could go to the gym.
When she goes "full" time for you, the rate can always be negotiated. Make sure you have a clear agreement. Also, you said it is his girlfriend...how stable are they? Do you like her more than your brother? If they break it off, you are going to be without childcare and in more of their life than you may want!