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.
Step 4: A Quick Fix
Step 5: Adding Tension
This part may be a bit different for you, depending on how big of a runner band you are using. Essentially, we want a lot of tension on the rubber band when the actuator is pressed. To create this tension, stretch the rubber band around the gear pegs in the bell. For even more tension, I pulled the loop part of the knot itself around the center post as well.
With the rubber band wound, place the actuator back on its pivot peg with the slot over the center post. The rubber band will pull the actuator up against the center post. Test this a few times to see if the band pulls the actuator back in place. You may need to wrap it around the posts another time.
With that in place, carefully put the gears back in place. The small toothed gear goes on the right peg first. Then the piece holding the washers goes on the center peg. Finally, the nut can be put back on top to hold it all down.
Step 6: Test It Out
Obviously, the rubber band will eventually break, but this quick fix has lasted for a few weeks now on my wife's bike and sure beats buying another bell. When the rubber band does break, it can easily be replaced in the same manner it was originally put in place. Maybe some day, I will come up with a more permanent solution...