As amusing as it would have been, Mechanical Package was not a song sung to the tune of Detachable Penis. It was actually magazine that ran from 1931-1932. Each of its four issues included a kit to build something such an an electric motor. The first issue, available online, included the article, "A Modernistic Den for Bachelors:"
The Editors of the Mechanical Package Magazine frankly admire the new style of contemporary interior decorating, and have kindly commissioned me to design a series of men’s rooms which the man who has the creative ability can build for himself at a very low cost.
And from the I-didn’t-know-that-had-a-name department, here’s the TV Tropes Wiki.