Tom,
The rectangular unit on the coil with the element in it is the starting resistor. They are notorious for breaking down. They are however relatively cheap to replace.
The coil is designed to operate at a voltage less than 12 volts. Ususally 9-10V. The resitor drops a couple of volts across it during normal operation, so that the coil gets the required 9-10V. During starting when cranking voltage can be less than 12 volts, the resistor is shorted out by the ignition keyswitch in the start position. This gives full battery voltage under cranking conditions to the coil.
Temporarily short ot the resistor and take the car for a run. If that fixes your problems, then replace the resistor.