How to replace pull-switch on ceiling light?
Honestly that looks like a fairly inexpensive light fixture. Why not consider just purchasing a replacement fixture?

A fixture built with the riveted type of wire connectors, ceramic bulb socket and with that type of sleeving on the wires is a high temperature rated fixture. It will really not be feasible to try to repair the unit and keep it's rating intact.

Get yourself a replacement fixture with LED lighting to reduce power consumption and lower overall operating temperature.