Since there was no erratic behavior of any sort, I figured this was just a faulty discrete component somewhere and set out to look for it. If certain parts of your IPC are not illuminated when they should be, there is a good chance that this is what's wrong with it. I have found lots of people who had or are having the exact same issue, and this procedure fixed them all!
Step 1: Helpful Tools
To remove and open the instrument panel cluster I used:
Socket Set with Extension - Primarily a 7mm Socket
Battery Terminal Wrench - Equivalent to a 1/4" socket in most cases
Auto Carpet Plug Puller - Pliers and a flat head screwdriver should work as well
Flat head screwdriver
To fix the IPC I used:
Soldering iron and solder
4 x 150 ohm resistors
Step 2: Don't Get Killed
Step 3: Dismantling the Dashboard
To remove the IPC, you will pretty much need to dismantle the entire dash compartment. Most of the plastic pieces are slid into place on one end and bolted at the other. Removing the bolts allows you to slide the pieces out. The other thing in the way is all of the carpet pieces used to cover up the mess of wires below the dash. These can be tricky to remove with a plug puller, but you should be able to pry them out with other tools.
A good idea is to start with the small pieces on the sides of the dash (what would be up against the inside of the car doors when they are closed, and work towards the large dash pieces from there. The screws and plugs will all be on the under side of the dash, so get down there and get to it! Try to only remove what you have to in order to get to the IPC, and don't lose any of those screws...
Step 4: IPC Removal
With the harness unhooked, there should be nothing stopping the IPC from coming out. Be sure to put the car back in park so it doesn't roll off!
Step 5: Open It Up
Step 6: The Fix
With the case completely removed, you should be able to see the complete circuit board. Locate a group of four resistors in the upper right hand corner of the circuit board. They will not look like typical resistors, but more like metal cylinders. You may notice that one or more of these resistors is crumbling or has come off of the board completely - this is the reason the circuit isn't working correctly.
Remove the resistors and clean off the pads with rubbing alcohol. Then, add a drop of solder to each pad to prep it for the new resistors.
You will need four 150 ohm resistors (1/4 Watt) to replace the old ones. Through hole resistors (Color code: brown - green - brown - gold/silver) are fine. Clip the leads and fold them over before coating each lead with a bit of solder. Then, attach the resistors one by one where the old ones used to be. Don't worry if you accidentally bridge them together (like I did), they are all in parallel anyway.
When you are finished, you can use a multimeter to check your work. Four parallel 150 ohm resistors should have an equivalent resistance of 37.5 ohms.
When you are satisfied, put the panel back together and put it back in the car. You may want to test it out before you put all of the dash back together as well. Don't forget to reconnect the battery cable! If it still doesn't work right, then there is probably something else more seriously wrong.