Looks like the drive is indeed dead - you might like to try it on another PC if you can, just to be sure. Edit: scratch that, it’s a laptop drive
I have no knowledge of replacement laptop drives (desktop user here), but hopefully other members can suggest a suitable replacement for you.
Another (easier!) option, would be just to buy a new external DVD burner, which plugs into you laptop’s USB port (assuming your lappy supports USB2.0) - LG make very good external DVD burners.