If you want a cheaper hotel room, the Cabinn Metro is 2 stops from the airport on the train (therefore about 20 minutes journey time including waiting) and far less than DKK2000, their rooms are in the DKK600 range. I do not recommend the Copenhagen Go hotel, even though it looks closer, because the rooms are small, old, and crappy.

If you still want to sleep in the airport, !'s description above seems quite complete.