I figured it out. I was poking around and saw that somebody else had an issue with the electrolytic capacitor and how it had lifted off the board. I re-soldered the capacitor down onto the board and thought I fixed it. It seemed to work, but once the board heated up from the residual heat I was still having issues.
I decided to look at the IR LED's through my phone camera, and only one was lighting up. As I pressed on the unlit LED, it lit up. I re-soldered the unlit LED, and now they both light up, and the LED doesn't lift off the broken solder joint as the board heats up.
If anybody else out there has issues with their boards and wants to check that the LED is working, look through your phone camera in the dark. It should show up as a pink/purple light on the screen.