I have tried offering the King the Clan invite when I first encountered him, after I completed the Mines, and after I picked up the Dragon Armor. He won't accept it. I was able to give it to one of his guards but he just says that it is a tempting offer but he is dedicated to the King. What I am I doing wrong?

You did it correctly. The King's not interested, but the guards "will consider the offer." When you return to the Guild, you'll get credit for the quest.

-Sazerac