PSA: If the Dagannoth Kings CA "From One King to Another" seems bugged and isn't completing, stop attacking after using your Rune Thrownaxe special attack.

I completed the task like six times and it wasn't registering, so I looked around and saw reports of people having it be inconsistent in prior leagues as well. It seems like the issue occurs when you continue to attack after launching the spec that kills Prime; it needs to be your most recent attack when the death animation finishes. If anyone else is having trouble like I was, try clicking under your character after the spec and see if that fixes it. (Make sure auto-retaliate is off, too.) The task works both in the normal and private caves. Hope that helps!