Comment: As pointed out in this documentary, it's impossible to separate "Holy Land" archaeology from contemporary politics, but the question of whether King David's empire ever existed is interesting in and of itself. In fact, since we're talking about Bible history here, it's a valid question to ask if David ever existed at all. As usual, however, there's no way to answer these questions with any certainty. Digging into the history just leads you into a bottomless quagmire of claims and counter-claims and controversial evidence.
"To most Israelis it is axiomatic that the celebrations for the 3,000th anniversary of the conquest of Jerusalem by King David mark a real and tangible event; but this is far from certain. The biblical account of the capture of the city is the only one we have, and in the opinion of most modern scholars, the Bible is not an entirely reliable historical document. Corroborating evidence is required, and some indeed exists; but it is not conclusive. When all the available information has been assembled, the most that can be said is that there was probably an Israelite ruler called David, who made Jerusalem his capital sometime in the tenth century BCE. However, the precise date cannot be determined, and consequently there is no way of knowing exactly when the anniversary falls." Source: Jewish Virtual Library.
There appears to be an ongoing controversy about whether any government or kingdom even existed in the region at the time. Some hard evidence has been discovered, however. For instance, "Six clay seals found at the archaeological site of Khirbet Summeily in Israel offer evidence that supports the existence of Biblical Kings David and Solomon," according to a 2014 article in Science News. These clay seals or bullae are thought to have been used "to seal official correspondence in much the same way wax seals were used on official documents in later periods," implying that some kind of government existed.
"The impressions in the bullae do not contain writing," however, so even if they are official seals, they don't tell us anything about who controlled the government at the time or how large it was. In other words, despite what the Science News article says, they don't constitute evidence that King David actually existed.
More direct evidence has been discovered, but it seems to be pretty sketchy. Some archaeologists believe that an inscription found on the Tel Dan Stele refers to the "House of David," but the accuracy of this translation has apparently been questioned. According to Kenneth Humphreys, "...this interpretation of the fragments has been challenged, both by a realignment of the 3 fragments [of the stele] and a corrected rendering of the word "BYTDWD" – not 'House of David' but a place-name meaning 'House of Praise'".
Humphreys is a well-known proponent of the theory that Jesus was a mythical figure. According to him, the King David described in the Old Testament is just as legendary, arguing that if David actually existed, he was little more than a bandit:
"The ancient Jewish hero David, 'son of Jesse the Bethlehemite,' is a 'curiously elusive figure' (Oxford Companion to the Bible). In turns shepherd, giant killer, court musician, poet, warlord and king, nothing and no one outside the Bible notes his existence.
"If a historical 'David' inspired tales of a legendary king, the original was an inconsequential bandit chieftain in the Judaean hills, nothing more. Possibly the only element of truth in the biblical story is the episode of David as renegade and outlaw leader, living from theft."
So what's the truth? Did King David exist? Did he rule an empire? The answer is that we don't know. There just isn't enough hard archaeological evidence to say one way or another, and, like usual, the absence of evidence isn't evidence of absence. It's possible that David is a real historical figure heavily mythologized in the Bible, but it's also possible -- probable in my opinion -- that he is a mythological character inserted into the real history of the Middle East, much like Sir Arthur Conan Doyle placed Sherlock Holmes in Victorian England.