Many options. Here's a few thoughts.
The first option that came to my mind was looking for accomodation in the Christianshavn area. Still in central Copenhagen though, but a different area to Frederikstaden where Phoenix Hotel is located. And a direct and easy trip with the metro to the airport.
There's also Dragør. A charming old village literally on the far side of the airport, and with fairly frequent bus service to the airport. Don't expect too much action, still it's a popular destination in the summer. Could be a nice little place to spend a final night in Denmark.
Yet another option is the town of Roskilde, about 25 minutes from central Copenhagen on the frequent trains. There are direct trains from Copenhagen to the airport once an hour on weekdays (you have to travel via Copenhagen Central on weekends). The "problem" is you risk not having enough time in Roskilde. The two main sights in Roskilde, the viking ship museum and cathedral, are well worth a visit - but could also be reached as a day trip during your 6 night stay in Copenhagen.