Special name for royal titles?

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.

You seem to looking for a sobriquet, an epithet, or a moniker.