This is my go to hotel in Copenhagen whenever possible.
While SP34 is not the most affordable, it offers up a comfortable lobby space that invites you to spend time there and a selection of onsite restaurants and bars that will let you feel like you went out even though you are only minutes from bed. Perfect for those jet lagged first days in a city.
The several stays I've had here have been an absolute pleasure. The SP34 is located in the old Latin Quarter, near Copenhagen City Hall and about a 10-15 minute walk from Copenhagen Central Station (København V). The neighbourhood itself is one of the original Bohemian enclaves in the city and has continued to live up to its eclectic roots with interesting bespoke shops, little bodegas, and interesting cafes abound.
Things to Do
Rent a bike
at the hotel or down the street at Sogreni for a much cooler/less touristy model
Next Door Cafe
I love this little spot around the corner on Larsbjørnsstræde. Great oatmeal bowls and baking for the morning.
Copenhagen's busiest market is within 10 minutes on foot (if that) and 5 by bike.