Iceland was part of Denmark from it's conquest until it's independence in 1918, however it still was under danish influence until WWII when it was occupied by the UK after the nazis occupied Denmark and in 1944 it became a republic, a status that continues until this day.
This makes Iceland the only nordic nation that is a republic, but what if it was different? What if Iceland went the same way as norway and after breaking free they became a kingdom? Who could be a king? In OTL the first kingdom of modern norway, Haakon VII, was born in Copenhagen as Prince Carl of Denmark, so could Denmark provide a King for Iceland? Also would Iceland be called a Kingdom or due it's size it could have another name like "Principality"?