Why doesn't a doorbell work after a power outage?
What kind of meter are you using? To check the wiring turn the power off
First check the button.  measure the button if not lit it should be open or very high resistance, when pressed very low resistance (ohms) But you should see a change in either case.

I would then take the chime directly across the transformer. If it doesn't ring my guess would be the transformer as a 16vac transformer unloaded would normally be closer to 20v rms. 

Next check the wires, you have a line connected to the transformer and a line connected to the chime, remove one of those wires and measure the resistance (ohms scale) from the wire removed to the other it should be a high value close to infinite (unless it is lighted button then remove it and retest) next short the doorbell wires and repeat the resistance measurement now it should be low probably less than 20 ohms if these 2 measurements are correct the button wiring is fine not shorted or open. 

The last item is the chime itself but I think you tested it.