If I jump the solenoid by connecting the positive battery to the starter motor stud on the solenoid... the engine cranks. However, if I connect the positive battery to the ignition stud (purple wire) on the solenoid... nothing. Any ideas?

Test the solenoid wire with a test light. Turn key to start it should light.
If so and engine doesn't crank replace solenoid.
If it doesn't light check safety switches.
Luck

Hi,
Sounds as if you have correctly tasted the solenoid and it is bad.
Peace,
Clarke