Danhostel Copenhagen Amager puts you within a 10-minute drive of Tivoli Gardens and Ny Carlsberg Glyptotek. Free perks include WiFi and self parking. Also, National Museum of Denmark and City Hall Square are within a short drive.
Pros: Decent distance to metro and airport. 10min walking and 20min train/metro to airport. Many showers.Cons: Too expensive for a hostel that's not in the center of things. Extra $7 for linens (no towel) is a bit excessive. Not enough electric outlets in room/next to beds, where you can charge your phone