Basically, you set up a business presence in the US (an address and a phone number.. which could be a paid virtual office service and a SkypeIn number... and stick with US bank accounts, file a D/B/A, etc.), then you simply work remotely, all the time.
From the Danish perspective -- my experience is in France, but I know other European countries have variations on the same theme -- basically you prove to them that you have sufficient money and external income to stay in Denmark, housing (your gf would sign an affidavit), health insurance, and they give you a visa for a year, which you can renew each year by proving the same thing over again.
In France you have to apply from outside the country, so if you have to get things set up in the US this would be the time to fly back there to get the process going.
Whatever you do, best of luck; my wife & I have worked our way through different immigration statuses in various countries, and it's always frustrating.