You should also check the starter actuation wire, as I call it.. That will cause the problem youre referring to. Sounds like your other cables/links are fine. It may have come off.
It is the small wire that goes onto a male spade connector on the top of the starter, check it.

 

I'll have my dad do that when he gets home. I just had toe surgery and I am not allowed to be on my feet. It really sucks. To all that have contributed to this thread, I would like to say thank you for your input and all the solutions that it will hopefully bring.

 

If I'm not mistaken, that wire is what feeds the solenoid. If that wire were disconnected, I believe there'd be a whole lot of silence when the key was turned.

 

Yep.

Bottom line, You have a bad ground somewhere. Either the ground to the starter, or the ground to the engine is loose or corroded. I suggest cleaning both battery cables and the battery terminals with a wire brush and then checking your work on the starter. Have you or anyone else had the tranny out that you know of? Under the oil pedestal where the tranny and engine meet is where the ground strap should be, If not this could explain previous problems with the car starting. Sometimes people will attach this to the starter instead of the engine and maybe you didn't notice it when you pulled the old starter. Fwiw, I made this same mistake years ago replacing a starter and couldn't figure out why it wouldn't crank one day. Turned out the ground strap was disconnected.

 

We are testing the starter now, and find that the motor is working but somehow the solonoid is not sending the juice to the starter motor. What does the ground strap look like?

 

Normally it's just a regular wire that goes to the battery. It comes off the engine, Then it bolts to the body were some wire is stripped away and then goes to the battery. Where it bolts to the body is where it usually gets pretty corroded and can cause no starts. If you don't have that setup anymore than it should be just a wire or braided strap from the engine to the body. Either way you need a good engine ground.

 

Or you can always install a starter button, it will involve some rewireing but rotorhead shows you how to do it.

http://www.rotorhead.ca/howto_startbutton.php