You'll need to get residency first. They don't use work visas per say, it's the EU which regulates it all. I'd apply for residency first, even without a job. You can ask for residency based on you living with your girlfriend (EU laws are a lot more relaxed). If you were in Sweden I could help out more, but unfortunately I'm just across the water from CPH.

Denmark also has a central job site (run by the govt), and LinkedIn has a lot of jobs there as well.