After reading a few different translations on this chapter and thinking it over I cant say who he is refering with the king piece (no one is actually talking about this but is bothering me
).
He is talking about pieces like if they where people (nothing so abstract as feelings or ideals), he said that taking the game from the point of view of konoha. This clearly leaves the daymio out of the game cause he is not part of konoha (and even if it was involved I cant see how the daymio is more important to a hidden village than their kage).
With this reasoning we have discarded: daymio, feeling or ideal, hokage. Now this makes the naruto being the king more reasonable but still I cant understand how Naruto is so important. Even if he was the 4th son the kage title isnt inherited, and is the kyubii so important to konoha? Yes is a trump card that they can use but it can easily turn against them. Is like a weapon, something that is good to have but the most important thing of konoha??
I cant see it, if naruto is really what he is talking about I think that asuma is way more stupid than what I thought. So after all this senseless text im going to say that the king are the children of konoha, it would match with 3rd hokages feelings and make a bit more sense than a stupid kid with a monster inside that could destroy all your country be the most important thing to konoha.
Im sure that in the next chapter when akatsuki aim for asuma we will know what this king thing is.