The DVD drive just needs to be replaced. It's a specific model of LG drive inside of these JVC DVD recorders. Open the unit, read the model number, and then head to eBay.com
to see if one is available. You won't find a new drive that works, so don't even bother. You'll need to find a pre-owned LG drives for it. This DVD recorder is worth repairing, so don't give up on it. It may take a little bit of time to find the drive, checking 1-2 times per week, but eventually you'll get what you need for it.
(If you do decide to give up on the unit, post it for sale on this site's marketplace forum
. Somebody will want it.)
This could probably be used for parts: http://www.ebay.com/itm/JVC-DR-MV5S-...item27bf050a0f
$60 on eBay
This one is $20: http://www.ebay.com/itm/JVC-DR-M10S-...item20bd931d4b
It has a bad power supply, but no mention of the DVD burner being bad! It may be fine.