The type found in David and Solomon is exciting. David battles and fights but is forbidden from building the temple. Jesus won the victory over sin at His 1st coming, but He did not come to build a temple or establish a government. He is coming back to do both and reign as the King of Kings. His reign will be a reign of peace like Solomon's was. It will also be the reign of the wisest King of all, the most glorious King of all, and the King to whom all the people of the world come to learn from.
When He returns, he will have to deal first with a usurper that has tried to take His throne. This points to the destruction of the anti-christ. He will also deal with an errant priesthood. "So Solomon thrust out Abiathar from being priest unto the LORD; that he might fulfil the word of the LORD, which he spake concerning the house of Eli in Shiloh." 1 Kings 2:27 He will remove the priests that have denied His name and replace it as predicted. He will be the ultimate Prophet, Priest, and King all in one. A Priest/King after Melchisidec's order.
There is more, but I'll let you have the joy of seeing Jesus in the narrative of David and Solomon. As with any type, of course, the sinful actions of men have nothing to do with Jesus. That is the Holy Spirit using the author to give us an accurate historical narrative and not an embellished fairy tale.