Eilert,
Yep, unfortunately you'd need to do all 14 exams and the IR/GFT flight test. Although, and I'm sure you'll be overjoyed at this, you won't have to do the morse code exam. There is an exemption for those that have over 3000 hours PIC on "Large" aircraft, flying on international routes. But it limits you to only flying aircraft registered in the JAR country, in which the license is issued.
The other thing is that you really need EU citizenship. Just having a Danish residency permit/visa, will most likely not be enough. Before I got an EU passport, I had a UK residency permit but still had to get a visa for France. So you can see the issues that would be involved as a pilot.