1999 Saturn SW2 SW2 won't start

There are two similar questions already posted. Neither one gives a working answers.

As a rule, you open the door with a keyless remote, start the car and drive away. But, on occasion, everything appears to work but when you turn the key, nothing happens. It just won't turn over, or even act like it is trying too. After a while, if you let the car sit, it will start. Sometimes you can speed the process up by doing a Billy Gates. You know, open the door, slam the door, lock and unlock the doors a couple of times, close the window, open the window. Get in, get out. Bang on the lower side of the steering column. Try the ignition again. Sometimes it takes two or three random cycles to get it to work. We've noticed that the problem usually follows opening the driver door and then opening the hatch back with the remote. I think it has something to do with a 'kill' relay, except I don't know it the car even has a kill swtch built into the starting/alarm system.