Download the Opera installer, launch it, click "options", set "install path" to your existing Opera (should be "C:\Program Files\Opera") so that it says "upgrade" and set "install for" to "All users". Check "Make Opera my default browser" and uncheck any options you want and then install.

That should fix your installation of Opera where it should then show up in Default Programs in the Control Panel. Choose "Set default by program" and give Opera all of its associations.