I really don't think that you'll be able to get any kind of plausible government from post-Revolutionary America where the man in charge has the actual title of "King". The whole point of the American Revolution (at least in the minds of the Revolutionaries) was to throw off the tyrannical monarchy and gain their full rights as Englishmen. At some point the full rights and being Englishmen became mutually exclusive, so they wanted their full rights as men, period. The Revolution was really the Americans revolting against a rather outrageously stupid Parliament (I would argue that the American Revolution was one most tragic episode in modern history, just imagine what Britain and America have both achieved separately, and then add it together, and presto, you have a [mostly] benevolent world empire).
So then I think that you could have the Constitutional Convention of '87 fall apart, a further spate of uprisings by farmers (think multiple Shays' Rebellions) and then when the next Constitutional Convention meets (say in '91) the specter of the further rebellions, combined with the increasing social unrest in France (which most of the American ruling class viewed as a bad thing) means that Madison's push for a more unitary state, along with Hamilton's vision of a more authoritarian republic (I think that is the right phrase for his views) would result in a life-time presidency for George Washington. So he'll die sometime before or at his OTL death, and after that, I think that you may have civil war. Hamilton will be much more active in crushing opposition and imposing his vision on America, Jefferson will probably be even more romantically attached to his vision of the yeoman farmer, and the anti-French feelings will stoked to the point of war. So perhaps Washington has a life-time appointment, followed by Hamilton or Hamilton's puppet, perhaps a civil war, more rule by Hamilton, then he dies and there is another constitutional convention, or another civil war or something.