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.
My favorite gallery in Copenhagen, located in the meatpacking area... its great for spotting up and coming talents but also established artists from around the world.
Nestled just north of the city, Dyrehaven is a vast, surreal landscape of meadows, castles, and gnarled fairy-tale like forests — Inhabited by a seemingly infinite population of wild deer. Lovely for runs, strolls, woodland adventures, and the like.