Here a switch has been added to control an existing receptacle. The switch takes the hot from the middle receptacles. With this arrangement, all the other receptacles are wired directly to the hot source and the switch controls the middle outlet.
This wiring illustrates the switched-receptacle circuit with the switch coming first. The hot source is wired to the bottom terminal on the switch and the top terminal connects to the black running to the receptacle. The source neutral is spliced through to the receptacle neutral.
Here a 2-way push-button switch is wired to a lamp with 2 bulbs. This diagram can be used to rewire an old push-button lamp with a new switch replacement. The hot wire from the cord is connected directly to the black wire on the switch and the neutral is spliced to the neutral contact on each bulb sockets. The red and blue wires from the switch are each connected to the hot contact on one of the bulb sockets.
The battery is the weakest link here and should be the first place to look for trouble. The battery can be replaced with a large flashlight-type dry cell rated at 12 to 16 volts and if the wires are still intact and the contacts clean, the doorbell should come back to life.