Douglas Alpenstock wrote:as a supplement to a mechanical connection -- not a replacement.
I wanted to highlight this part of what you wrote. Yes. First you repair the connection mechanically, then you use the solder to ensure a reliable electrical connection between two things. Trying to use solder to provide strength for the connection will not provide for a durable repair.
You may be able to find a new plug, or adapt an IEC standard plug (those ubiquitous 3-prong ones that they used to use on things like computers) if the opening in the case is of an appropriate size. The latter approach would require a new cord, although it is a common type.