I think you're thinking of Solomon. I don't know of any archaeological evidence for David, but Solomon's cylinder seals have been found and portions of his palace--which makes him about as well-attested as Gilgamesh.Mythological leaders and kingdoms? Honestly, this strikes me as an odd thing to say, as there is archaeological evidence to support that king David existed, and the kingdom of Israel, besides being mentioned by the 1st Century historian Flavius Josephus, and in the Talmud. That is pretty convincing, no? When I think of mythological, I think of characters like Thor and Odin, not like king David.
Also, while the Bible was never intended to be a history book per se, I don't see the need to examine it more critically than other ancient sources when it discusses history, particularly of the monarchic period. Maybe we can talk about the Patriarchs being of dubious historicity, but no one is suggesting Abraham lead Israel.