The pull chain broke off the switch in my closet ceiling light. I bought a replacement switch, and was going to replace it, but the lead connected to the light fixture appears to be riveted to the terminal, and I can't figure out how to free the current wire.
I also thought of just replacing the pull chain itself (since the switch is still good), but the case appears to be riveted together, and I can't get to the chain fragment inside.
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.
Orange wire nuts. Yellow will also work. Get the ones with metal inserts. Strip the wires 3/8", align the ends and twist until things firm up.
Michael Karas mentions the heat rating of the fixture. He's right, but very few situations call for that level of protection. Use LED bulbs or stay below 60W and you're not going to have a problem. Put the fiberglass mat between the bulbs and the wiring if you want a little peace of mind.
Disclaimer: I can't possibly be 100% sure you won't burn your house down, but this is what I'd do in my own home if I had reason to not replace the fixture.
