Is there a name for the title/nickname that some members of royalty get? For example, "King Larry the Kind".
Epithet is the closest in one of its meanings precisely matching "the Kind" (or "the Conqueror", "the Bastard", "the Great" for well-known historical case). Unfortunately, a more recent sense focusing on derogatory terms is increasingly the better known, to the point where it might seem wrong to some readers or listeners who don't know the earlier use.
Cognomen though that's also used of general nicknames, and part of the historical Roman naming system (the term's origin).
Sobriquet if it replaces their name (so Caligua would be an example).
Edit: Some time later, but I've suddenly just remembered the word byname, while having more general senses, is uses for the sort of "X the Y" name that distinguished them from other monarchs, etc. of the same name.
"the Kind" the "Great" are epithets, that is correct. Just to add something to this discussion:
Style refers to the address a particular member of the aristocracy should be attributed to. For example, the Queen gets a style of "Her Majesty" and other members of the Royal Family gets "His/Her Royal Highness".
Also, there are several ways to style a prince or a princess.For example, the Prince of Wales is styled as 'HRH The Prince Charles, Prince of Wales'. On the other hand, the Princess Royal is styled as HRH The Princess Anne, Princess Royal. As a son of the English Monarch, a royal prince gets to have THE prefixed on his name. Compare this with HRH Prince William, Duke of Cambridge, and HRH Prince Henry of Wales. Obviously, as soon as their father, THE Prince Charles, assumes the throne, they will be styled, respectively as HRH THE Prince Charles, Duke of Cambridge, and HRH THE Prince Henry.