Sounds like you either have dust or dirt in your headphone jack, which can be removed with a thin object such as an unfolded paperclip, or you just need a replacement. The headphone jack is part of the speaker housing assembly on the SIII. It's a very easy fix. About 8 screws or so, can't remember exactly, located behind the back plastic cover. Just unplug the old and plug in the new. Part isn't very expensive either I don't believe. Can be found on repairseunivers.com
It could always just be a defective cable. This might sound like a silly question but have you turned the phone off and back on? Sometimes powering the device off and back on is required to detect a new cable that has recently been plugged in. If that's not the case I would imagine it's just a bad cable. I would try another one.
What you are talking about it actually in the backlight. There is a couple different options for fixing it. The first option would be to completely remove that backlight, cable and all from that assembly and solder on another, iPhone 5 backlight might work, not sure. Your other option, if soldering is out of the question, would be to remove the film and filters from an iPhone 5 backlight and place on the 5c LED strip of the 5c backlight. Be careful with the LCD tho, expensive stuff.