Starter not engaging

Battery drained, bad brake switch. Replaced the switch button only and started car but did not tighten battery posts. Car would not start an hour later. Appeared to be a dead battery. Recharged, tired to jump, tried known good battery. Starter would just click once and that is it. Put in new starter, same thing. Turn on ignition and hear one small click and nothing else. Now this is the interesting part: Turn ignition on all dash light come on, door chime works. Turn to start all light go off except the engine light, wait twenty seconds and it all comes back on, but still will never turn the starter.