Perhaps you have heard of the Kingdom of Gwynedd in Wales?One can assume that the stories about David and Solomon, no different from the reports about Abraham and Mose, are only edification and propaganda literature, albeit on a high narrative level. They are part of Deuteronomistic history (the books of Joshua, Judge, Samuel and Kings) and were written to illustrate the consequences of obedience and disobedience to the laws listed in Deuteronomy (5Mose) using David and Solomon as examples. Solomon, as he is portrayed in the Tanach, is a mythical figure and Jerusalem, according to archaeologists Finkelstein/Silberman, was only a village of 1000 inhabitants in the 10th century BCE. All information about Solomon is either fictitious or, if the myth about him has a true core, greatly exaggerated. The Kings Book claims that he had hundreds of wives and concubines. That would have been the entire population of Jerusalem. And how would a village chief finance this harem?
Wales has an area of 20,779 square kilometers or 8,023 square miles. On maps, it looks like the Kingdom of Gwynedd had an area of about a quarter of Wales, and thus about 5,194 square kilometers or about 2,000 square miles. It looks to me like the United Kingdom of Israel and Judah under Solomon would have been about half the size of Wales and thus about twice the size of Gwynedd according to the biblical narrative. But Solomon's kingdom, if it existed, might have been much smaller, and thus possibly about the size of the Kingdom of Gwynedd.
The main royal residence of the Kings of Gwynedd was at or near the village of Aberffraw on the island of Anglesey from about 860 to 1170. From what I read there is little archaeological evidence of a llyn or royal enclosure there, though history says there was one there. The kings of Gwynedd may have also been the high kings of all the kingdoms of the Romano-Britons during the post Roman age and on into the Middle Ages. Being "the King of the Britons" would make the King of Gwynedd the British equivalent of the High King of Ireland. The High King of Ireland theoretically had his "capital" at Tara Hill, and often had the title of "King of Tara". And perhaps due to the same way of thinking, the King of Gwynedd and King of the Britons was often called "The King of Aberffraw", though his realm was many, many times as large as that village.
So calling Solomon "The King of Jerusalem" need not imply that that his realm was restricted to a few square miles around Jerusalem, especially when it is remembered that medieval rulers using the title of "King of Jerusalem" ruled an area about as large as the Bible claims that Solomon's kingdom was.
And it is still the practice among some English speakers to call the ruler of a large realm, many times as large as Solomon's could have been, the ruler of his capital city. It is common to call the ruler of the Roman Empire "The Emperor of Rome", and write of "The Latin Emperor of Constantinople", and speak of mighty Muslim monarchs as the "The Caliph of Cordoba", "The Caliph of Baghdad", or the "Sultan of Delhi".
So as far as I can see, there is no strong evidence to prove whether Solomon's kingdom occupied zero square miles or kilometers, or thousands of them, or was in between in size.
And you may ask what sort of archaeological remains have been found of the courts and capitals of the King of Kings of Ethiopia, a realm many times greater than Solomon's, from the year 1600 or the year 1500, or the year 1400?
No archaeological remains have been found as far as I know, and I don't expect any could ever be found. From what I heard, in those centuries the King of Kings of Ethiopia and his court constantly traveled from one place to another, forming a vast tent city whenever they stayed in a location. So unless someone knows they camped near a particular village, town, or city, and knows a logical camping ground, there would be no place to dig for traces of the court. And Ethiopia was so powerful that over a thousand years earlier, the prophet Mani had said that the four great powers in the world were Rome, Persia, China, and Aksum or Axum, the precursor of Ethiopia.