Jesus says that David speaks through Holy Spirit the phrase "Lord spoke to my Lord: seat at the right hand of me"; thus, David asserts that the one to whom Lord spoke those words is his, David's Lord and therefore, Jesus says that David called Messiah his (David's) Lord. An analogy: for example if David were a schoolboy and his math teacher would say to his literature teacher: "this guy is not good in math, so perhaps you make a poet out of him", and David would have said: "Teacher (of math) told my teacher (of literature): 'make this boy a poet, for he has no perspective in math".
Thus, I do not see what is your problem in this? It is as clear as a sunny day: David, in Jesus' words, addresses to his Lord (the messiah) to whom Lord (God) has spoken. By this, Jesus simply underscores His divinity, for if a prophet-King speaks of Him as "Lord", He cannot be just another king of Israel like David, even if He were an eschatological king-messiah who would have restored the Jewish kingdom, because even such an eschatological king would not posit himself higher than David, but respect the latter as the great ancestral prophet-king putting him even higher than himself, as a son (descendent) would put a father (ancestor), for this is what Jews thought about the Messiah - "the son (i.e. descendant) of David" only in a physical sense. And even if - let's hypothesize - such an eschatological Messiah-king would not posit David above himself, then at least would not posit him below himself either, to be sure.
On the contrary, Jesus clearly and unequivocally posits Himself not only just higher than David, but proclaims that He is no lesser Lord for David than Jahve Himself, and this David said being educated by the Holy Spirit, for it is only through the Holy Spirit that someone can discern the divinity, the Jahve-hood of Jesus, as Paul says (1 Cor. 12:3).
Thus, Jesus deconstructs the basic expectation of Jews, that Messianic king will be somebody like David, fulfilling the desire of a political grandeur of Jewish nation, but Jesus' and the Father's Kingdom is not of this earth (John 18:36), not of political dimension, but within human hearts (Luke 17:21) that are cleansed from the infection of sins through the action of divine grace working through faith (Eph. 2:8).