Do you *ever* need to use "of" with possessive "s"?

"An agent of King Ronald's" is completely correct, as any grammar book will tell you. It is called a "double genitive" and just like "A friend of mine," "A friend of his," etc. are grammatically correct, so too is "An agent of King Ronald's". No doubt some poorly educated teachers have mistakenly taught otherwise but that happens a lot.

The Chicago Manual of Style: The possessive form may be preceded by of where one of several is implied. “A friend of Dick’s” and “a friend of his” are equally acceptable. 7.29

The AP Style Book (DOUBLE POSSESSIVE): Two conditions must apply for a double possessive–a phrase such as a friend of John’s–to occur:

• the word after of must refer to an animate object, and
• the word before of must involve only a portion of the animate object’s possessions.

Otherwise, do not use the possessive form of the word after of: The friends of John Adams mourned his death. (All the friends were involved.) He is a friend of the college. (Not college’s because college is inanimate).

Merriam-Webster's Concise Dictionary of English Usage:

[The double genitive] ... is an idiomatic construction of long standing in English -- going back before Chaucer's time -- and should be of little interest except to learners of the language, because, as far as we know, it gives native speakers no trouble whatsoever.

But the double genitive was discovered by the 18th-century grammarians and has consequently been the subject of considerable speculation, explanation, and sometimes disapproving comment.

The double genitive is standard English and should not be worried about.

Cambridge Grammar of the English Language: Chapter 5 "Nouns and noun phrases",

§16.5.3, "Alternating patterns of complementation":

• Mary's green eyes / those green eyes of Mary's
• Mary's book / that book of Mary's
• Mary's secretary / that secretary of Mary's
• Mary's new house / that new house of Mary's

You don't have to do it, but it is common to say (if not write) "a friend of Ronald's", by analogy with "a friend of mine". "A friend of me" might be logically defensible but idiomatically, it is emphatically wrong.

The structure is called the "double genitive".

Also, if King Ronald owned many paintings, including one depicting his sister, and she owned one depicting him, there would be "a portrait of King Ronald's" and "a portrait of King Ronald".