Once you've beat the game, you'll be able to access the Dragovian Sanctuary. Defeat the Lord of the Dragovians there, and once he's defeated, you will be able to participate in the Dragovian Trials. You'll be forced to fight the Lord of the Dragovians again at this point. Once you've defeated him for a second time, request the Dragovian Sword.

From here, you should talk to the woman at the nearby item shop, being sure you're behind the counter to talk to her. She'll give you a recipe that will allow you to create the Dragovian King Sword. Thereafter, simply mix the Dragovian Sword you won earlier with the Liquid Metal Sword, and you will get the Dragovian King Sword.