They honored him enough to bury his body because he was the descendant of Jehoshaphat, who was a good king. This can be stated explicitly. Alternate translation: they buried his corpse, because they said, "He deserves to be buried, because he was a descendant of Jehoshaphat, who tried hard to please Yahweh." (See: [[rc://en/ta/man/jit/figs-explicit]])
Here "house of Ahaziah" is a metonym that refers to Ahaziah's descendants. Here "power to rule" is a metonym for a king. Since all the adult descendants of Ahaziah were dead, there was no one left to rule Judah. (See: [[rc://en/ta/man/jit/figs-metonymy]])