From what I can see, it’s most likely the card has failed (e.g. dried up solder joint, bad capacitor, etc.). I had an nVidia FX 5700 AGP fail in the same way about two years ago, but for me the PC was working one day and then didn’t boot up the next and discovered it was the card when I tried another AGP card I had handy.
It’s probably cheaper to get a new VGA card of a similar type than to get it repaired. If you use eBay, give it a try, as you’ll likely find one there.