You might wanna try Lagkagehuset A/S.

This is a bakery shop with a cafe that sells all sorts of sweets and cookies.

And there is also an old cady factory with their own shop right in the center of Copenhagen called Sømods Bolcher that you might like.

And for good vegetarian food in a cosy location I would recommend Atlas Bar. They have a good vegetarian menu and good informal service.

Lagkagehuset A/S is definitely a standard, and luckily somewhat ubiquitous.  For your sweet obsession, I'd also suggest Meyers Bagerimy personal favorite bakery anywhere, and Mirabelle which was recently opened by Christian Puglisi. The baker there, Carol, is someone I briefly worked with and is a force to be reckoned with!  Once you're there I'm sure she can tell you all about her favorite pastry shops in the area.

Also on the ice cream front there's Østerberg Ice Cream which I've heard great things about, along with Isobar which was a regular spot for us while we were there.

Lastly, I hope you do have some fine dining plans while in Copenhagen...the dessert ethos over there is unique and definitely eye-opening.

Bertels Kager Salon http://bertelskager.dk - excellent pastry shop. You won't regret it.