A friend and I want to spend a week in Copenhagen to see the city. Copenhagen is ridiculously expensive and I am searching for a place to stay for both of us without bringing a sleeping bag with us. Airbnb is a hassle so I would prefer something else.
The cheapest hotel that I'm aware of in Copenhagen (you can of course find much cheaper hostels) where you have your own private room and bathroom is Cabinn.
They have several locations in Copenhagen, I prefer the one near the main train station, "Kobenhavn City Hotel" as they call it.
The rooms are small, but clean and reasonably comfortable.
The cheapest price for a two person room is 625 DKK total (i.e. for both guests), assuming they are not sold out for your dates.
While you may occasionally find as cheap and, more rarely cheaper rooms elsewhere, I've found Cabinn to be consistently right at the bottom range for hotels. I haven't stayed anywhere else in Copenhagen since 2005.
The cheapest way to stay in Copenhagen (or, indeed, in any other city) would be by Couchsurfing.