Whenever I feel uninspired I visit the (now genericly named) Design Museum, formerly known as Kunstindustrimuseet. Sort of the V&A of Copenhagen. An old museum filled with gorgeus fabrics, weird artifacts and a cozy library filled with books about arts & crafts from around the world. If you're still uninspired after a visit here you're not able to. Don't forget to have lunch in the beautiful center garden surrounding by the overgrown buildings, statues and old trees.
Ordrupgaard is a beautiful museum with lots of interesting exhibitions. Right next to the museum Finn jul's house is located, a famous danish designer and architect.
Just 25 min by train from downtown Copenhagen lies an old and beautiful forest and country area inhabited by wild (but friendly) deer. It's a vast, mysterious space for getting lost in and perfect for spacing out on mushrooms if you bring friends. Stay away on Sundays as it fills up with screaming children and zombie parents. Grab an ice cream on your way back at Bakken, the old David Lynch-ish fairground located at the outskirts of the forest.