So it's been a couple of weeks since this begun, but at the moment, the headphone jack on the front of my PC is void. I don't yet own an audio interface, but my big plan is to buy one this year (new year, new gear) for my studio use and for the family, who are also keen on using headphones. Are there any solutions to this problem? Maybe something cheap or something I can have my hands on quickly or possibly already own?

There's usually a headphone jack on the back of the card where the expansions slots are.
Did you look there for one?
 

Line 6 has a good single jack portable interface that comes with the pod station amp sim.

Look used, I'm sure you'll be able to find something under 100 that will be a HUGE step up from the built in sound card.

If this is a laptop, I doubt if it's repairable. There is a slim possibility that you might be able to re-route the jacks and use the input jack as an output. My laptop has Bluetooth so I could use that. 
 
If this is a desktop then I would put a Cheap PCI sound card in it. But that's $20 you could put towards a real interface. 

Dumb question - you sure it is the jack and not the headphone cable, plug or wire?????

Good question, actually. I've seen many cases of small diameter wires breaking inside the casing, just behind the boot. It's a very common failure in light duty headphones and power supplies. Much more common than a failure of the jack itself.

Depending on he PC brand, many have a "daughter board" with the I/O jacks on that and it can be replaced.  If its actually the jack that is bad, look into whether yours can be replaced. 
 
Otherwise it may just be a driver corruption and reinstalling the correct driver (the PC vendor's driver is preferred) may be all that's needed. 

for the time being, I've found the solution of plugging a BOSS BR-532 PDS into the line-out of my desktop. The only pair of headphones that have successfully worked with the headphone jack is a pair of Sony MDR-7506's, which even had a little bit of trouble find the 'sweet-spot' inside the jack. I'm thinking of getting a headphone amp, but I'd hate to cut into my sloooooowly growing audio interface budget, since that'll eventually replace the amp anyway. I'll try some contact cleaner in it, but I don't think i'll even be reverting to this output (since the BOSS PDS can amp up a quite bit more than the headphone jack).

Fix it, Jack!!!