Oooh an American Witan! Very nice. There is still going to be an interest in invigorating the union's legislature, and the legislature will have an acrimonious relationship with the King for a few decades.
The cultural differences, and genealogical differences between the citizenry of the different colonies had become quite pronounced by the time of the Revolution; and this was reinforced by the different economic models and religious denominations of the colonies. A monarch of the union would have been an emperor to start with, especially one that came from a foreign people.
To create a uniform nation the king would have had to attempt to suppress the cultural differences and foster a uniformity. I do not believe such a task is possible for America by that late of date.
Alternatively, the king could rule explicitly as an emperor. Not a lot of people got on board with the idea of the nationalists (that Americans were a single nation, versus > 13 nations) when the Constitution was being promoted. The Federalists had a lot of popular opinion trouble with their more centralized view of governance. If France has its revolution, that will be the ultimate test for a monarch of the United States. As an aside, I think the French Revolution could have been defused almost up to the last minute - but most people here probably do not agree with me on that.
The advantage that the King will have is that the length of his tenure will let him have a somewhat more stabilizing affect on policy. Some actions he could take to increase his victory is spearheading the negotiations for Vermont's entry, the Eerie Triangle, admitting Tennessee and Kentucky, and adding a super West Virginia (maybe with Pittsburgh). Getting to the XYZ affair should improve his reputation. Henry Louis might be able to speed up the negotiation of the Louisiana Purchase, and it would be somewhat mythically sentimental if the news of France's agreement gets back around the time he dies.