The genius of this song never ends.  But your question has debatable answers depending on your interpretation of a true haiku.
Each verse is just slightly off of the traditional 17 syllable English-written haiku by one or two syllables except the verse in the middle,
"La Da De Diddum La Da Da Dum Do Diddle Dum Do La De Do"
which satisfies the definition of a haiku.
Additionally, the verse Joe sings is cleverly written because he's messing up a rhyme while telling about how he is no good at rhyming:
"I've never thought much Of formulaic verse anyway And rhymes are not my fort(e)"
Good stuff.  Probably one of the most cleverly written songs I've ever known of and surely one of my favorite Tally Hall songs.
Edit:  It's since been pointed out that "fort" is an acceptable alternative  pronunciation  of forte.