The ignition switch looks like the likely culprit, but to tell if it's the relay or the solenoid, can you have someone else turn the key for you while you listen from the front of the car with the hood up? It should then be obvious if it's the solenoid clicking or not. But then, that's just a symptom of a discharged battery, and doesn't really explain why the battery is not holding a charge. If you have a voltmeter, you might want to check the voltage across the battery terminals -- should go up to around 14V when the engine is running (once you get it going), if not suspect the alternator. (Oh, I see you did check the alternator, but what did you do to eliminate it?)

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'81 GLT 245 W/245K; '83 DL 175K