Most times, doorbell problems are button, especially since indicator light shows you have power.
Take off front doorbell button and see two wires.
These wires are low voltage and safe to touch.
Use screwdriver and touch both screws together, or remove wires from button and then touch two wires together.
If doorbell rings, then replace button.
If bell still doesn't work then scratch two wires together and look for static-type spark that says wires have power.
If wires have power, and you crossed both wires and bell didn't work, then new door chime is suspect.
If wires do not have power, then transformer is suspect.
Open following link to see example transformer:http://waterheatertimer.org/images/attic-transformer10.jpg
Transformers are located in basement or attic.
Scratch a wire across both screws on transformer and look for static-like spark.
If no power, then check circuit breaker.
Home centers sell transformers in the doorbell section. They are 12 and 24 volt. Check markings on old transformer.
If transformer has power, then new door chime is suspect.
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