We'll have to beg to differ on our opinions of this.
I think the rather comedic feel to "Up and down the goddamn town"
sits rather nicely with the almost jovial "......But I'm your own light-hearted friend
Jack the Ripper"
It would be nice to find the source of the first two verses or to lay eyes on the original that Mcnaghten cited, though as with much of Jack the Ripper history there is little substantiated, it's always just beyond our grasp.
‘There is nothing more deceptive than an obvious fact’ Sherlock Holmes