There is another Input jack on the back of the unit, but the two jacks are wired up such that the jack with the broken tip in it has priority.
This is a diagram of the front panel input connector. Basically, pin 1 is the tip of the cable (signal) and pin 3 is the sleeve (ground). When no cable is plugged in, pin 2 connects to pin 1, and when the cable is plugged in pin 1 disconnects from pin 2. The back panel input jack is wired to pin 2. This means that when a cable is plugged into the secondary input, the unit will use that signal, but if a cable is plugged into the main input, the secondary is disconnected and only the main signal will be used.
This works great, until a cable tip gets stuck in the main input jack, rendering it useless, and also effectively disabling the secondary input.
Begin the tear down!
Oh, I guess I'm not supposed to remove case...
Or... maybe I will...
Well... there's your problem. But how in the heck to I get it out?
Ahhh... it appears the front panel is just a facade.
Alligator clips for the win
Just when I was going to give up, I noticed these...
Fuses!
The picture doesn't really do it justice, but the two 0.75A fuses were very milky white... and I couldn't see if the fuse wire was still intact. I pulled out the trusty multimeter and it turns out both of them are blown. Something must have shorted in this delay unit, and hopefully it wasn't me. Either way, I'm off to buy some fuses.