Harry has the Night Watch regularly, but not nightly.
We can look at at TNG and modern naval practices can give some idea of what a watch rotation might look like.
In TNG 6x10: Chain of Command, Part 1, we learn that the Enterprise operates on three duty shifts when Captain Jellico changes it to four. The Enterprise crew, or at least Riker, is resistant to the change. But honestly, there are people in the real world that believe a shorter work day is actually more productiveA,B,C - Jellico may have been on to something.
If Voyager uses three watches, like Enterprise, then poor Harry may be pulling sixteen hour days - a pace he probably can't keep up indefinitely. If Voyager uses four watches, like Jellico prefers, that's only twelve hour days - rough, but more reasonable over a longer period of time.
VOY 7x08: Nightingale provides a bit of evidence that Voyager may use shorter watch rotations than Enterprise:
JANEWAY: You've been in command on Voyager before.
KIM: On the night shift, for a few hours. It's not the same thing. You and Chakotay are always a couple of decks away, ready to take over if anything goes wrong. I've been on Voyager for almost seven years and I'm still an Ensign.
From http://www.chakoteya.net/Voyager/708.htm, emphasis added.
In VOY 7x19: Author, Author, Harry tells his parents he only gets to do it twice a week:
MARY KIM [on screen]: I was hoping you could record a short presentation about what it's like to command a starship.
KIM: But I don't command a starship.
MARY KIM [on screen]: In your letters you said the captain puts you in charge sometimes.
KIM: What I said was I'm in command of the night shift, twice a week.
From http://www.chakoteya.net/Voyager/717.htm
In the real world, there are a wide variety of watchkeeping rotations, some of which divide the duties into even smaller slices than 6 or 8 hours. If Voyager uses one of these real-world schemes, his "night watch" may only be an extra four hours. That said, if he has a night watch and a day watch, that probably makes him an afternoon sleeper or perhaps one of those people that can successfully and easily pull off polyphasic sleep.
Again though, there isn't really enough information to know for sure.