Jing has a point. If the king is "your precious person" then everyone is knight and king at the same time.

So yeah, the king must be an actual, tangible thing - either a person (most plausible thing) or a place. The Adjective, too, has a point - nobody would bother to keep such a thing secret.

I think that whetever the King piece is it has to do with what Asuma wanted to tell Kakashi but was never able to. Now, if we think of some other big-ass secret that hasn't been revealed, which comes to mind?

The only one I can think of is what Sasuke finds out about his clan after Itachi has murdered all. It might be a stretch, but they can be connected - after all, the Uchiha were said to be Konoha's "police force", which is in charge of protecting the village... just like a knight.