It is certainly a lot easier to get married but as you both have decided against it for now, you would need to either apply for a EEA Family permit from the BHC in Denmark (which is valid for six moths) or enter the UK as a non visa national. Once you're e in the UK you need to apply for a UK residence card as an unmarried partner of a EEA citizen. The EU Directive considers unmarried partners as extended family members, as such the rights of residence of your EEA partner are not automatically conferred on you. Also the directive states that matters of residence for non-EEA unmarried partners be decided by domestic legislation (in your case UK Immigration rules). So be armed with as much evidence of your long term co habitation as possibe (at least 2 years worth). The law is not very clear on the status of non-EEA unmarried partners while their application for a residence card is being considered. So I would advise that you may find it a bit of a challenge to seek employment, open a bank, register with the NHS while you wait for your RC to b e granted.

I'd advise you to do as much research as possible from now, also maybe contact independent immigration advisory organisations for advice.
Also see http://www.bia.homeoffice.gov.uk/eucitizens

Good luck with it