I had similar issues with au pairs - rarely got evening babysitting out of them in the 4 years we had them. I think I got 2 date nights with DH, and possibly 1 night of my own (he was away 15 nights, consecutive, out of every 28 at the time).
In my case, while I desperately wanted some free time, the most important thing was that au pair could do mornings and get primary aged DD up and out to school, and preferably collect her twice a week from after school club. I was usually there when DD woke, but had to leave to get to work over an hour before school gates opened (walking distance).
So I picked my battles - if AP was happy, met friends in her English classes, and settled so would stay the contract (or longer in a couple of cases), that was good. But I did have 2 that didn’t settle, 1 we were able to work around getting a replacement before she left, but 1 dropped on us, the night after her DSis had spent 4 nights in our house and the evening before DH went away again, that she wanted another family straight away. That was a Saturday night and she was gone the next Friday evening. (I managed to find a different solution that then worked for 2 more years, but no thanks to that AP and led to a very very stressful month for me).
So make sure AP knows what you expect, agree now what Saturday night and a few other babysitting times, you want for the next few weeks. But also keep her happy with her friends etc so that she will stay the contract.