Copenhagen hotel and area

We stayed 2 weeks ago in Absalon Hotel--1 1/2 blocks from the train station. You ride the train from the airport to the Central Station.

Absalon's breakfast was the best I've had in any hotel anywhere. And our room had been completely redone recently--including new plumbing. It's a very popular place at a fair price.
I would say Copenhagen is a better walking city than it is a good public transit city. Most tourists can walk to the palaces and popular tourist sights from the Absalon.

The area around the Central Station has many budget to mid range hotels, which is the case for most European cities. However the area behind the Central Station on Istedgade also has the 'Red light district', while nearby Vesterbrogade is much nicer. Though I am sure David can attest that it's not that bad.
Generally speaking Copenhagnen is very walkable and while public transport is good, it is aslo expensive, especially for short rides, were you might as well walk.
I can't comment on particular hotels, since i've never really needed one.

For such a lengthy stay have you considered an apartment? You will feel more like a local and spend so much less time and money on food, and it is wonderfully relaxing to eat take out every so often. If you stay out of center you will better appreciate the great rail, metro, and bus systems. Hotel and restaurant costs are the only things that can make Copenhagen feel expensive. English skills are so high even way out of the tourist areas you will be just fine.

My wife and I just got back from Copenhagen on Monday. We stayed at the Hotel Kong Arthur for seven nights. The hotel is just off of the Norre Sogade, very close to one of the canals. It is only a five minute walk to the Norreport Station (which has direct trains to just about everywhere on Zealand). We thought the hotel was very convenient. It was about a 20 minute walk to the Central train station. We walked to most of the sites (for reference, it was maybe a 20 minute walk to Christiansborg Slot, and maybe 30 minutes to Christianshavn), but you can easily take public transportation to these locations from Norreport. The hotel itself was reasonably priced, very clean, very nice. We did not eat breakfast there (it was expensive and not included in our package, although it is for some), but saw it and it looked delicious. They have a cozy hour too, where you get a free beer/glass of wine/non-alcoholic drink, and if you book directly with their website, you have free access to their spa (we didn't use this).

I would definitely recommend this hotel.

Thank you everyone! I will take a look at all of your suggestions.