Article: Leverage open the tabs along the switch using a screwdriver. Once you separate it into its 2 pieces, the things inside of it, like the chain and spring, will be visible. With the switch open, remove it from the wiring by pulling the wires out from the switch. Be careful not to lose any of the pieces when you pop the switch open. If you don't see the spring and chain, pop off the protective plate holding them inside the switch. Remove the contents of the switch, including the broken piece of chain connected to a disc and the spring. All of the contents inside the switch will be loose, making it easy to simply take them out. Pop the broken chain off of the disc using a screwdriver. Discard the broken chain after you've removed it. Just as you removed the short broken chain, pop a longer chain that will reach through the fixture into place in the old chain's spot. Snap the longer chain into the disc where the broken chain was using your fingers. Put the spring and chain attached to the disc back into the switch. Pull the chain through the hole in the switch so you can pull on it easily. If you popped off a protective plate that holds everything together, press this back into place too. Hold down the spring and chain mechanism using your finger or a screwdriver so it doesn't pop up while you're attaching the two main switch pieces together. The 2 switch halves will press back into each other easily. Put the switch back in the wires just as you took it out. Make sure you place the switch next to the spot where you unscrewed the nut from the outside of the fixture. Screw the nut back into place to hold the switch.

What is a summary?
Pop open the switch using a small screwdriver. Remove the broken piece of chain from the mechanism. Put a longer piece of chain in the old chain's spot. Reassemble the switch by putting the chain and spring back into place. Place the switch back inside the fixture and reattach the switch nut.