This will probably be moved elsewhere but I will try to give you some insider knowledge. Since I don't know when you plan to visit Copenhagen, I can't give you much info on events relating to fashion because there is fashion/design week also.
So there are some brands based in Copenhagen but I have no idea how easy it would be to contact them for a studio visit. There are Cecilie Bahnsen, Ganni, Stine Goya, Henrik Vibskov, Heliot Emil, division, Helmstedt, Baum und Pferdgarten, Saks Potts, Nicklas Skovgaard and so on. It depends a little what type of design we are talking also. There is also Andersen Andersen who make beautiful quality knitwear for example but it's not groundbreaking design.
The Design museum could be interesting, as well as A. Petersen as they often have textile exhibitions. You could also be lucky with Kvadrat showing a fashion-related exhibition in their showroom but that is maybe not too likely. GL Strand sometimes shows exhibitions with fashion photographers or at least I recall one by Mario Testino.
For stores I would say Storm, Birger Christensen, Illum, studio x, Mr Larkin, Holly Golightly, Acne Archive, Paloma Vintage, Jeromes, Times Up Fashion. Apollo will sometimes host flea markets where the Copenhagen cool kids will sell out of their closets.
I think it really depends what your aim with this trip is and what sort of fashion design you are interested in. I would be happy to send you more brands/designers that are smaller but based in Copenhagen.
