I believe both the main and low cost terminals at KUL are open 24 hours. I can't spot anything on the LCCT terminal website indicating otherwise. (The main terminal is definitely 24 hours)
For an overnight layover like that, you might want to consider just buying access to the Plaza Premium Lounge in the LCCT. That'll get you a shower, some food and drink, wifi (normally better than the airport wifi), and a comfy seat to snooze in. I've not used the LCCT one, but I've seen lots of people doing that same thing in the one in the main lounge!
If you do decide to buy access (assuming you don't get free access via one of the few airport lounge access programs, or some higher end credit cards), it's worth booking online in advance. You can normally manage a decent discount over the walk-in prices.