How do you repair a pull chain that has broken off inside the fan

First, remove from fan, sit in area where you won't lose small part(s), You will see if it is doable for yourself, Rivets are toughest, can be done. Have chain ready to install.
Home depot has some switches. There are times, switches need to be repaired because of availability.

Replace the switch. See:

Ceiling fan pull chain replacement and repair - Ceiling Fans N More (http://www.ceiling-fans-n-more.com/ceiling-fan-pullchain-replacement-and-repair.php)