What's the distance between the computer and the TV? Like networking cables, there is a maximum distance that can be used depending on various factors such as cabling quality and surrounding "electrical noise". For less than 20 feets, any digital cabling (DVI or HDMI) will be able to carry the signal with little to no quality loss, even the cheap <10$ ones you find on the web.
Unfortunately, with a DVI to HDMI conversion, there is no way to have sound, you need to carry the sound cable separately. Some new cards have HDMI built-in and it is possible to plug the sound card in them to send sound+video over the HDMI cable.
If you have 2 DVI ports, yes you can plug one in the TV and the other in the monitor. Then you can either use split-screen display (half the display on one and half on the other) or mirroring (same display on both).
Would this one do the job? (sound+video, and good quality)
Component video cable also only carries images (see wikipedia if you want more info). Component cables can support up to 1080p, but it is an analog signal and therefore more susceptible to cable quality and electrical noise. Moreover, not all TVs support 1080p input through component video cable AND the blue-ray standard offer a way for manufacturer to downgrade quality for non-encrypted signal (like component), so the quality could vary from 1 movie to another if they use that feature.
Ok, it wasn't easy, but I was able to find the kind of connector the TV has (I don't know all the names, I only know most by sight).
Not all is maybe you should contact LG help line or the manual to make sure I don't say anything stupid.
First, I recommend using the a digital medium for the image (DVI to HDMI cable or DVI to HDMI dongle + HDMI cable), much less quality loss due to the cabling and everything, you should be able to find something fairly cheap online. This one should be plugged in HDMI port 1 of your TV, which is marked for DVI (on the brochure I found). You should then be able to plug your soundcard's output directly into your TV (just below the VGA connector marked RGB) using a male/male mini stereo jack (2nd connector from the right on this picture).
Just look behind your TV and tell me if my instructions were clear enough.
If you don't want to risk anything, try to go buy the necessary cables at BestBuy, Wall Mart or any large store with a reliable return policy (but much greater prices) and if it does the trick, return them and go buy your own online at a fair price.