I agree with the other user comments here on this site that state it helps to like Danny Kaye in the first place, because the film offers nothing fresh and exciting outside of a love for musicals and Kaye's effervescent madcap malarkey. It's a perfect showcase for Kaye to let loose and he delivers smartly as the humble milkman mistakenly built up as a prize fighter of note who then proceeds to lose the grip on his ego. He is surrounded by very stoic actors and they all benefit from a tidy script and foot tapping tunes, and sure enough the laughs are dotted throughout the show, but it still feels like they plonked Danny Kaye on set and built a film around him.
It's also of interest to note the back story of the film actually being a remake of Harold Lloyd's 1936 film The Milky Way, that is something that few people are aware of and great effort was made by the makers of The Kid From Brooklyn to distance themselves from the 36 film. So with that in mind it's hard to not view this film as merely a Kaye vehicle without much heart, and with that I say the film is entertaining enough without being close to being a really good Danny Kaye movie, 6/10.