There are a number of smaller hotels near Nyhavn (north end of the Stroget) that are worth checking out. July is actually the best time to book hotels in Copenhagen, because most of them cater to business conferences so summer is actually their downtime.
The three hotels I've been to in order of niceness are: The Phoenix, very good service and one of the best breakfasts anywhere in Europe, just a couple blocks from Amalienborg palace; The Strand, about four of five blocks from Nyhavn; and The Opera, just around the corner from the Kongens Nytorv and Nyhavn (this one's a little threadbare but otherwise nice). None of them are modern but they are all decent accommodations with good staff.
All three are walking distance from the metro stop in Kongens Nytorv, which is an incredibly easy ride directly from the airport (maybe 15 minutes, faster than the train from the airport into central station).
I really like the area around Nyhavn, it's quieter and cleaner than the area around the train station, city hall and Tivoli - but you can still walk down the Stroget to get to that area.