From the history recited and the fact that the country has a name, then "king" should be correct. In the Chinese traditional understanding of statecraft, the Empire was nameless. It was simply "the nation", "the middle kingdom", or "tian xia (all under heaven)". Dynasties would come and give it their names, but it would be understood as "Great [Dynasty name]", for example "Da Ming" instead of "the country of [name of region]", for example "Qin Guo". So, based on the naming of the state, it should be a kingdom ruled by a king.