There are currently no sleeping places* The airport and the passenger hotel mentioned by Mark are only accessible to you if you are traveling for more than 8 hours. You will also need a special pass and a special bus to get you there as it is in a high security area.
There are three lounges at the airport: the VIP Lounge, the Pearl Lounge and the Emirates Lounge.
The Pearl Lounge offers chargeable facilities. You only find it near the immigration desk. There are no sleeping areas, but there is a kid-friendly area, good food, free Wi-Fi and comfortable chairs and is a good place to spend time.
The Emirates Lounge should be avoided best. It is just a secluded area with poor food.
The VIP Lounge is not open to the public.
Apart from that, there are no special transit facilities.
The good news is that there are very good catering options and the airport is open 24 hours and has relatively little traffic at night. Since the terminal is quite small, you should rest near the food lounge. There is a new area from which some Asian airlines depart and this will be relatively quiet.
I described the transport experience here and here. In summary, it is best to avoid having to renovate the airport.
The airport is currently being extensively refurbished with new terminals and a redesigned main terminal. One of them is dedicated to Kuwait Airways (Terminal 4) and is due to open in August 2018 for commercial traffic. Jazeera Airways (the local LCC) has its own terminal, called Terminal 5, which is already in operation.
There is also the GA terminal (called Sheikh Saad Terminal) for Flydubai. and finally the New The airport, called Terminal 2, which is still under construction, will be the main hub.