Malmö is just across the bridge. Quite a few people working in Copenhagen live here. Not that it is much easier to find a place here...|
As it happens a former colleague is going to Japan soon for some unspecific amount of time and is looking to rent out is flat. If Malmö would work for you and if you are not looking for something permanent I could ask him if he has found someone yet. If I remember correctly he is asking for around 7000 SEK/month.
#10237510, By Syrok Finding somewhere to live in Copenhagen