I haven't ever stayed there, but yes, it is inside the airport! For such a late flight, it would be very convenient. Plus I'm sure you could get room service if you're hungry.
We recently were stuck at the airport for 3 extra nights (made the best of it though!!), and got rooms at the Rennaissance right nearby. I have to say that we were very very happy with our rooms (2 queens connecting to a king) but my husband is Marriott platinum, so we got special treatment (club level rooms, and their club level was very nice). They do have a shuttle, which I think they share with the Courtyard (which didn't look half as nice; I had to go over there to use their laundry room). So if for some reason you can't get the Hyatt Regency, I do recommend the Renn. For just a night, and the convenience, I'd do the Hyatt.
You can check out tripadvisor for reviews.
