I have a similar schedule to you, often up until 3 or 4
I find I feel alot better if I work out, so often will go at 2 am, but I prefer to go around midnight if I can, but going at 2 am is better than not going at all
sometimes the night janitors are busy cleaning but no problem, it is a nice atmosphere actually when there are only a dozen people in the gym in the middle of the night