Our philosophy is "you wouldn't wake a sleeping baby, so don't wake the sleeping grandma!" We do need to adjust her schedule at times, like getting up in time for church and appts, but generally, I feel very blessed that my mother-in-law likes to sleep. I get myself up and together early enough that I have a chance to lie back down for a bit before I get her up, which I need for my back problem. That gets me through pretty well if she goes down for a nap in the late afternoon for an hour or so.
I wouldn't worry too much about how long she's at the day center. I'm in the same boat with that pay by the day, not by the hour thing. I'm not worrying about it. I want her to get as much time there as she can, but not at the expense of our sanity! We'll get her there as early as we can, but we're going to be relaxed about it so she can enjoy her time there and I can enjoy her time not here.