i think that king has some kind of ability that makes him avoid taking damage, maybe awakening.
the flames are not a problem for zoro, and it's not haki or physical abilities, it's something more specific.
the only time king took damage is when marco attacked him, and we know that marco has special flames that can damage big mom's homies that cannot be damaged by traditional ways.
zoro has enough AP to cut kaido, but so far he didn't even scratch king, which is odd.
special damage techniques:
1-zoro's flying dragon blaze that killed a zombie without salt
2-brook's soul king DF ability
3-marco's phoenix flames