The office of President was completely invented by the Founders. There was no real precedent for a president (ho HO!) in any previous culture; Rome had two consuls, Athens had either a supreme assembly, or an individual who tyrannically consolidated power.
The most likely way to establish a monarchy would have been for the Convention to decide that the president's term was unlimited. A president for life, effectively an elected monarch. This was not unprecedented, at the time of the revolution Stanislaw II was the elected monarch of Poland. He was from a prominent noble family and elected king in 1764 at age 32. Poland had had a succession of elected kings over the last ~300 years, with monarchs coming from major families in Poland and Germany as well as relatives of the Swedish, Hungarian, and French kings.
To make this politically feasible, the elected King would have to meet the requirements of the constitution for a president (35 years old, natural born citizen) and would have the same powers (veto, commander-in-chief, appoint members of the cabinet and judiciary with consent of Senate, sign foreign treaties and declare war with the consent of the Senate). Since the cabinet would do most of the day to day business, which is of course beneath the dignity of the king, the king could appoint a prime minster, who would replace the vice president and be in charge of the cabinet. The Secretary of State, Defense, Treasury, Attorney General, etc would then work for the Prime minister, who is appointed by the King.
Washington would have (obviously) been the first king. On his death in 1799, there would be an election with John Adams and Thomas Jefferson as leading candidates. Since Jefferson won this election in real life in 1800, he would have been king until his death in 1826. In as much as John Adams' son was elected president in 1824, he would be a good bet to be the third king from 1826-1848. After that you have to figure it out (Zachary Taylor? Millard Filmore? Henry Clay? Winfield Scott? Alternate history is fun!)