Step 1: Get Familiar!
It seems there is some money available or a possible repair to those with this issue that meet the qualifications listed. If you don't meet the qualifications, you're not interested in the red tape or just like the satisfaction of fixing things your self continue you reading and I will show you how to fix your TV for under 10$!
Some tools you'll need going forward are a screwdriver, a soldering gun and some solder. Time to get this ship sailing!
Step 2: Open Up the TV
Step 3: Disconnect the Power Board
Before we get at our board make sure your TV is unplugged. Even once unplugged the TV can hold a lethal amount of energy in some of the bigger capacitors so press the on/off switch on your TV a couple times to help discharge some of that. Even still, be very careful with what you touch paying special attention to the HOT section of the board and all big mad scientist looking components, especially if you don't know what they are or what they do.
Now that you've got an idea of what you're looking at go ahead and unplug all the wires connected to the power board. Each wire group should have a different number of headers and should only fit back into the slot it came from but take note of where they did come out of so you don't have to fumble the board when you're putting it back in later. Next unscrew the board and lift it out.
Set the board on a nonconductive surface and we'll check it out!
Step 4: Remove the Bad Capacitors
If you're a super safety freak you may want to discharge these capacitors before you try to take them out. You can do this by touching the ends of a resistor between the two ends of each capacitor on the bottom of the board. But unless you have a massive capacitor that is your problem child it really shouldn't be an issue.
Before you remove the capacitor identify first identify its polarity in the circuit. The capacitor should have a white stripe up the side, note where this is as you will want to put in the replacements with the white stripe on the same side.
To remove the capacitors first identify the where the leads attach to the circuit at the bottom of the board. Warm up your soldering gun and heat the contact point on the circuit while gentle pulling the capacitor out of the board. Don't be too forceful pulling it out and don't hold the soldering gun at the contact for too long, both could damage your TV. Just be patient and you'll get it out. Remove all the bad capacitors.
Step 5: Replacing the Bad Capacitors
Place the capacitor back into the circuit remembering to orient the capacitor so it is has the correct polarity. Bend the leads out so it will stay in place as you solder. If you've never soldered before peep an instructable for some handy tips! Solder the sucker in place and be sure that your solder is not touching any other connection on the board (you don't want any shorts!). Repeat for each capacitor you need to replace.