The problem with modern archaeology and conflicts with Biblical history have more to do with the assumptions of early Egyptologists than with lack of archaeological evidence for King Saul and David. Modern archaeologists are looking in the right place but in the WRONG TIME.
King Saul, Jonathon, David, and the events depicted in the Bible are also recounted in the archaeogical record IF one looks with an understanding of the basic error made very early in Egyptology. That error was to equate Ramses the Great, Ramses II, with the pharoah of the Exodus. This error displaced Biblical and Egyptian synchronicity a couple of hundred years out of sync. It would be akin to looking for US vs. Soviet events among the records of the Revolutionary War.
The Amarna letters are a vast library of clay tablets that were written to Pharoah reporting on events on the Northern frontier of Egypt. Some of these letters from ambassadors and allied rulers describe the very same events the Bible reports including the rebelling of David, the slaying of Jonathon, the death of Saul, using contemporary appelations for these persons. The evidence of the Amarna letters has been dismissed by orthodox archaeologists because they DO NOT FIT THE ACCEPTED EGYPTIAN TIME LINE that was established by that original error.
Have you read David Rohl's excllent work "Pharoahs and Kings"? I highly recommend it. He lays out an alternative Egyptian timeline, documenting his evidence, that more accurately fits OTHER non-bibilical events, including records of solar and lunar eclipses, Pharoanic records, and dynastic histories, than does the accepted "orthodox" timeline based on the Ramses II/Moses connection. Rohl's timeline places the Kingdoms of Israel in the era where Egyptian hegemony was at its weakest: the time when Pharoah Ahkenaten (Born Amenhotep) was leading a religious revolution against the old gods declaring the ascendency of the god Aten (Sol) and ending shortly after the death of the boy Pharoah Tutankhenamen (born Tutankhenaten).
Ahkenaten's religious revolution resulted in chaos and unrest in Egypt. A power vacuum developed on Egypt's frontiers as political and military attention was turned inward. This was paralled by similar problems in Babylon, the other super-power of the era, and their attention also turned away from the Levant. Only in this hiatus period where the two ancient "big bullies" were otherwise occupied could the nation of Israel rise up to any semblence of prominence without being slapped down by one of the contending empires.
Once we accept this new synchronicity, examining the Amarna letters reveals many Biblical personages BY NAME and the events described are found to be the same events recounted in the Bible from another viewpoint!