Fascinating discussion. If we were to go the way of having retired Presidents become King or in a King-like position, there are several options. 1. We could use something similar to the Late Ottoman succession, in which they would become King in the order of their presidencies, for life. For example today Jimmy Carter would be King, and when he dies it would go to George H.W. Bush, then Clinton, then Bush Jr and finally Obama. Or we could have something similar to Malaysia. Malaysia has a King that is elected from the Rulers of the nine Hereditary Malay States and reigns for five years. A person can only become King again after all rulers of the other states have also stood for election. So it could be that a system would develop that the King would be chosen from the living former Presidents. Each would serve a number of years then the next would be elected. Not very likely initially mind you but would be the best for the US in later years,
Next he have a hereditary Monarchy, either with the House of Washington or the House of Hohenzollern (the Prussian Royal House for those who don't know). If its Washington the matter of Succession comes up. Would the throne go to his step-grandson,George Washington Parke Custis, or his nephew and heir Bushrod Washington. At this point the throne could become semi-elective, with the Washington family providing candidates and being elected King for life, either by direct election/electoral collage or by congress.
The interesting question is how powerful would the monarchy be compared to OTL's presidency. Initially it would be weak because of Americans understandable distrust and reluctance on having a King, similar to how none of the Roman Emperor's called themselves Rex, even when it was obvious that there was a monarchy. But would the Crown grow more powerful like the Presidency did? Would there be a hereditary Peerage/aristocracy? or would there be a Life-peerage type? What would the title of the Monarch be? King, Emperor or something unique/less royal sounding, like Princeps or First Consul? Would there be a Crown estate, like the British had or would there be a civil list? How would an American Monarchy change European perception of the US?
Personally I can see a Monarchal US developing similar to the early Terran Hegemony, from the Battletech universe. In it, members of the reigning Cameron dynasty were nominated to serve as Director-General and would be elected by the people, for life. Also, the nobility titles were granted for life to individuals who had achieved special accomplishments and were recognized for them. People like scientists or inventors or possibly artists. I would assume that some of the founding fathers families would be granted hereditary titles in recognition for their actions in the Revolutionary war and subsequent years.