I recently spent a weekend in Copenhagen and - following the starting points given in this thread - had a pretty good time. (Thank you!)
Indeed the "massage clinics" proved to be a good tip.
The system is as follows: All these places usually have only one girl present (sometimes 2), unlike in other places in Europe where up to 10 girls are waiting and you can make your selection. So it's crucial to check the website of the place and their schedule ("Vagtplan") before you go, so you go at a time where there is a girl that you like. also, make sure to re-check directly before you go as the schedule might have changed.
Also I would recommend to reserve the girl you have picked, just call and tell them you want to see xyz at that and that time. Worked well for me.
The rates seem to be pretty standard at all these places, a quickie is 500 DKK (about 67 Euro or 82 USD), with extras costing more.
The best deal to me seemed to be the half hour at 1000 DKK (134 Euro / 164 USD) which has all standard stuff included, so you have some time to get to know each other, cuddle, and there will be no discussions about extras.
Anyway, did some research and found these places in the Copenhagen area to have some better-looking girls:
http://www.cluberotica.dk
http://www.amorpigerne.dk
http://www.sms-planet.dk
http://www.pussygalores.dk
http://www.fanny-hill.dk
you can look at
www.realsex.dk for more, just click on "Massageklinikker - K