For an overnight layover at CPH, you can beat the location of the Clarion Hotel. Though it was a bit of a trek, there's a walkway from the airport directly to the hotel lobby, so we didn't have to go out in the cold/rainy February weather at all.
The front desk staff was friendly and welcoming, and the room was very clean and comfortable. While the amenities were a bit pricey (as is Copenhagen overall - dinner at the CPH airport was the most expensive meal of our whole trip to Greece, and it included a ~$2 tap water...literally water from the sink in a glass), we were impressed by the overall look of the hotel, layout and fixtures of our room, and convenience. Though we didn't shell out ~$30 for breakfast, I'm sure that would've been delicious. We also didn't spring for the ~$8 water bottle in the mini bar, but as the staff said, the tap water is actually pretty great (and free from the sink, unlike at the airport, so drink up!).
Overall, we enjoyed our quick stay here, and by staying at this hotel, I enjoyed my (possibly only - a little out of my price range!) visit to Denmark!