I use wire-wrap wire to fix broken tracks, or to modify boards. It's very thin, and easy to solder because of the silver coating. It breaks easily so some sort of glue to hold the wires in place is advisable, I just use masking tape, sometimes.
First off, I'm going to presume that the TV is out of warranty. If it is still covered, your best bet is to call the warranty line and get it fixed that way.
It's hard to tell from your popped resistor photo, but is there a heat sink to the left of it in the image that is black? There might be something under there that blew out too.
As for DIYing the repair, you could try swapping out that through hole resistor that blew. However, while it's common for low quality capacitor to give out, resistors don't usually pop unless there is an over current situation, so there might be a larger issue at hand here. It also looks like there are 0 ohm and a 47K (? hard to read) ohm surface mount resistors in the damaged area on the back of the PCB that might need replacing.
If it's currently broken, there's no harm in attempting a repair. Find a friend who does electronics to help you source the parts, especially if you think you'll need to replace those surface mount resistors.
As for getting a replacement board, rather than having it replicated from scratch, you should be able to get a replacement board. The various boards in TVs are usually modular (power, tuner, etc), and that appears to be the power supply board. Try searching for "TVMAKE TVMODEL Replacement Power Supply Board" and see what comes up. I know the Vizio 50" plasmas have a nasty habit of blowing power supplies and thus they're pretty easy to find replacements for online.
Good luck and stay away from your boyfriend's books lest they burst into flames. ;)
Best Answer
It looks like that pad (and another one as well) is not connected to any trace on this side. If you can solder it on the other side it should be sufficient.
If you cannot access it on the other side then one simple solution would be to find where that trace goes and solder piece of wire between jack pin(s) and any of the other pads on same net.
This will probably not hold the jack in place though, so some epoxy can be used on the other side to fix it to PCB.