Solved! KR uses Flash cookies, NOT normal cookies. This caused me about an hour of confusion before I posted. So the solution is to find out where flash cookies are stored in your computer (depends on your operating system) and to copy them to the correct folder.
If you play KR on Armor Games the folder with the Flash cookies in it should be called kingdom-rush-12141.swf because this is the name of the game on the Armor Games server.
Hope it helps if you have the same problem.