A broken remote, lots of electronic tools, a lazy New Year's Day, and just enough knowledge to be dangerous - What did I have to loose? I first tried to replace the LED with one from another old remote. The LED looked different and didn't work. So, I took it back out. Next, I added some leads from the printed circuit board of the remote and connected the original LED to the new leads. After re-assembling the remote, it worked perfectly. All were amazed (including me, I have to admit)!
As I told my wife, "Now, this hobby is finally paying off." ;-)