The citizenM hotel has a prime location, right by Rådhuspladsen (city hall) and Tivoli. The rooms are small--like, basically just a bed and shower--but clean, modern, and comfortable. Outside of the rooms, the hotel offers several amenities like a bar, meeting/work space, and plenty of space to hang out. It's not necessarily "cheap," but it may be more affordable than other centrally located hotels. (I haven't personally stayed here, but know people who have and liked it.)