I have an aging nutone doorbell that suddenly stopped working. I don't know whether the switch on the door is broken or the doorbell is. I disconnected the wires from the switch and touched them together but nothing happened. How can I be sure the door bell is broken?

Sure it could be broken but lets see if its getting power first. Do you know where the little transformer is located and what circuit its attached?

You already determined it was not the switch. Reconnect the switch and have someone push it while standing at the chime. Hear a buzz or see the plunger(s) move? If so, try to clean the little round strike rods by pulling them out of the return springs and cleaning with mineral spirits. Get a qtip and clean the rod bore. Then, wipe a very thin coat of 3N1 oil on the rods and reassemble. If no buzz or movement, you need to find the transformer and test the output voltage with a VOM. If you get 12-24 volts, either the bell winding is shorted or you have a broken wire in the circuit. If all else fails, get a wireless unit at HD or Lowe's. $30, you are in business and no wires needed.

Sometime the transformer is in the wall behind the chime.

Hard to know where they are sometimes. Brian

Sometimes the coil of either the front door or backdoor may fail, if you only have 1 door, or don't mind the same chime for both doors, you may be able to use the working chime, common and front, or common and rear.