I found a brown wire inside that was broken in half. It was a VERY tiny wire, the smallest I've ever seen, but I managed to repair the wire, and now the controllers are working again. I also remembered that I could use a multi-meter to check all the other wires to make sure they were working.
The connectors are supposed to be the problem in most cases, and mine look a little tarnished. Cleaning them didn't help. eBay has some brand new ones for sale, that are made of a different metal that won't tarnish.
I'm going to try to get some of these and replace the existing pins on my NES and see if that fixes it.
http://www.ebay.com/itm/380852831039
Here's hoping...... I'll post back if I ever get it working. It would be nice to be able to fix it. That means maybe I learned something from the starter kit after all.