Sorry, I was more elaborate in a different troubleshooting post.
Anyway, what you see is the original wiring. I have not changed or
done anything yet. The kit instructions say to hook the 3 plugs up to
the appropriate terminals on the compressor.

http://hdsupply.com/webapp/wcs/store...tegoryId=10400

The above link is similar to the kit I have. The 3 wires at the bottom
hook up to the compressor (common/start/run). The black wires on top
(toward the left on the picture) should be connected to the "power
supply".

I am guessing in this case they would connect to the 2 black wires on
the left side of my diagram (as the kit replaces the relay and cap).
But I am not sure. And, if this is where they go, I am not sure which
one goes to which.

The black box on the bottom is actually on the power cord. It is not a
box, but just a rubberized "enlargement" of the cord. (Seems like some
type of splitter manufactured onto the cord.)

The broken wires/cracked wires are one and the same. The insulation is
so old and dry that not only did they crack when I was manipulating the
wires, but the wires themselves broke as well.

At the bottom of my diagram is the plug. So, from the plug you have
the power supply cord with a green ground coming off, then the square
part on the cord from which a black wire comes out (on the plug side)
and 2 black, 2 orange and a brown that come off on the opposite
(appliance) side as diagrammed.