We really loved out stay at Conni's place during our trip to Copenhagen.
Big Advantage #1: Very easy to get to from the airport. Copenhagen's Metro is clean and easy to use, and it's about 20 minutes from the airport to Conni's place. You can see the Metro stop half a block away from the apartment.
Big Advantage #2: Lots of space. After a week of traveling in other cities, we appreciated the chance to spread out a little--have breakfast at a nice table, read in the living room, and have some space in the bedroom.
Big Advantage #3: Great neighborhood. The apartment is just barely outside most tourist maps, but it's so easy to get into heart of the city by Metro (5 minutes to most stops of interest) or by foot (we ended up walking almost everywhere). But check out the local area if you have time. We loved getting drinks on the Islands Brygge harbor (less than a 10 minute walk), and there's an inexpensive decent takeout pizza place a block away (Da Misto) and a grocery story nearby that we used to save some money in an otherwise expensive city.
Otherwise, it's everything you'd want. It was very clean and easy to get to, and Conni was a fantastic host. Her communication was excellent, and she was super friendly. Couldn't recommend this place more highly.