This wiring diagram illustrates installing a 15 amp circuit breaker for a 120 volt circuit. The 14/2 awg cable for this circuit includes 2 conductors and 1 ground. A 15 amp circuit is usually used for wall receptacle outlets and light fixtures.
Here 3-wire cable serves a split receptacle. The bottom half of the receptacle is controlled by a switch and the top half is always-hot.
This page contains wiring diagrams for adding a new light fixture using an existing receptacle as the source. Diagrams include adding a light from a switched receptacle and tapping an always-hot outlet. Different wiring arrangements are included to allow for either the light or the switch to come first in the circuit.
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.