Eventually, I decided to open the bell to see how it works and hopefully fix it. While my solution may not work for the entire life of the bell, it is very simple and should provide plenty more "bring-brings" until I have to work on it again.
Step 1: Things You Will Need
A small, strong, rubber band.
A pair of pliers to help with dissassembly.
Step 2: Opening the Bell
Step 3: Internal Mechanisms
There is a step up gear ratio in use here. As the actuator moves, it turns a gear with few teeth multiple times. In turn, this gear turns the removed piece holding the washers. Thus, a little movement of the actuator causes a lot of movement with the washer piece.
The actuator should be pulled back to its starting position automatically. This is done by a pair of plastic arms connected to the actuator and a small ring placed in the center of the bell. After a lot of use, these plastic arms eventually broke away from the actuator, causing the "spring" action to no longer work.
I don't know of anyway to fix this piece that will not break again soon after.