Electronic – How to safely discharge this capacitor with broken wire

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. ;)

With small capacitors up to 1 mF, there is little to worry about. I suppose it's a good idea to make sure they are discharged before plugging them in where the voltage that could be on the cap could damage something, but this is something not generally worried about until you get to some real energies or high voltages.

For small electrolytic caps like what you are working with, just short them against something metal, like a bare component lead, metal chassis, or handy screw driver.

Don't waste brain cycles thinking about this for anything small enough to be a ceramic cap you can plug into a breadboard. By the time you plug it in, your fingers will have discharged it. Even if not, do the math. 1 µF at 10 V is only 50 µJ. Yes microJoules. Big deal.