Going back for a moment to 'The Unfortunate Rake'...

Lighter wrote: "St. James Infirmary" is only a very distant relative. The situation is barely comparable and the tune is entirely different. The main point of commonality may be the appearance of "St. James" in the name of the hospital.

Pamela has already partly answered this, but to me the most significant feature is in the funeral arrangements that she mentioned. Compare:

"Get six jolly fellows to carry my coffin
And six pretty maidens to bear up my pall"

with:

"Get six gamblers to carry my coffin
Six chorus girls to sing me a song"

That sequence of six men (carrying a coffin, in both) followed by six women has to be more than convergence.