The Byzantine diplomacy is such that outright conversion of the King and the nobility will not be a demand - but, if Jerusalem is found to be in a time of need or desperation, some demands will be made.
They will most likely be relative to the Patriarchy, as said above, one or more high-profile marriages between the high nobilities of the Empire and the Kingdom (as OTL) plus some trade deal. The status of Antioch will certainly be a matter, as the Emperor will most likely demand it to become a direct vassal of the Empire (which will probably be accepted).
The only way you can get a more Christian Orthodox KoJ is through immigration of traders, artisans and other professions from Constantinople to Jerusalem and Acre (and the reverse-Latins from the Levant, Arab Christians and maybe a few Muslims) going from the Crusader states to Constaninople and then come back -some of them having adopted Byzantine culture and maybe Orthodoxy. Then some of them or their descendants may become nobles, but none of them will become King, Prince or Count of one of the basic vassals of the King of Jerusalem.
Ofc, the above paragraph depends on having a strong Byzantine state which also protects (mainly through diplomacy) the Kingdom.