Light contact tab melted into plastic housing
You don't solder something onto the center contact. Multiple reasons, one being it's already damaged form too much heat, another being the fact that a lamp socket should be UL rated, and that goes out the window when you modify it.

You buy a new lamp socket (at your local hardware store, or lacking one, online) and replace the whole socket. This will require some minor disassembly of the lamp, and you should carefully examine the cord to see if you should be replacing that as well due to old-age issues when replacing the socket.