I don't have any personal experience with hotels in Copenhagen, but I know the city quite well.
Regarding highlights to visit it really depends on your interests. So it's (almost) impossible to give solid advice without knowing more about your interests. I would, however, recommend everyone to take a canal cruise on their first visit to Copenhagen.
I've made this short (and very generalized) introduction to the main districts of Copenhagen, which can help you get an idea of where to look for accommodation:
Indre By: City centre and the oldest and part of the city. Compact and very pedestrian-friendly. A lot of the sights, attractions and museums in Copenhagen are located in Indre By or within walking distance from Indre By. But it's expensive and the most touristic part of the city. No matter if you sleep here or not, you will visit Indre By (several times) while you're in Copenhagen.
Vesterbro: Hip "hipster" district, with a good variety of bars and restaurants. Home to Kødbyen (foodie and nightlife area) and to the (by international standards: very "tame") red-light district around Istedgade - still perfectly safe though.
Nørrebro: Multicultural and lively district. Some similarities to Vesterbro regarding a hipster-feel (especially the inner parts of Nørrebro).
Østerbro: Considered as the "upper-class" residential district of Copenhagen, with nice big parks. This district is a bit posh in some places, family-friendly and generally quieter than the other parts than the city.
Frederiksberg: Frederiksberg is its own municipality (so it's not technically part of Copenhagen, even though it's fully enclosed by Copenhagen). Mostly an upper-middle-class residential area. Family-friendly, lots of greenery and good parks. Somewhat similar to Østerbro.