Thanks to earlier posts by Travis, among others, I figured out my own solution.
Zero thanks to Microsoft for this, and a brickbat for the lack of a response at all.
Check to see if your application is listed under HKEY_CLASSES_ROOT\Applications
If it's there, it will be a key named for the primary executable (utorrent.exe, for example).
If it isn't there, which mine wasn't, you will have to create it. I recommend that you study the key structures for other applications in this section.
If you go postal on your registry, you will screw up your system. In that case you will have to wipe it and reinstall Windows. So don't do that. But don't be terrified of editing the registry either. The worst you can do is mess it up, which is what's already
wrong.
You'll need to create the subkeys that you see in other applications. Take a look at the entry for notepad.exe, and for some other non-Microsoft applications you've installed, to see how to set up the keys, values, pathnames and pass a parameter to your application.
You'll need to create Shell, Open and Command subkeys, and give a value to the default in Command.
When you've created this correctly, the application should now appear as one of the choices, (maybe even the only choice) when you attempt to associate an application with the filetype.