On my lamp, the problem was not the chain but in the switch itself. This made it necessary to replace the entire socket since the switch was riveted to its base. I made this video which shows how to replace the socket and switch:https://youtu.be/0fLnUPWzLeI

A very nice and clear explanation. Unfortunately the pull chain switch in my lap was riveted to the socket so I had to replace the entire socket. It really was easy though and required less steps than your switch replacement. Here’s a video that shows exactly what I did: https://youtu.be/0fLnUPWzLeI

Thanks!