Attempts to reseat the graphics card didn't have any affect and upon closer inspection, it was clear the graphics card was dead:

As you can see from the image of the card in question, the capacitors have clearly "popped"; their tops are bulging outwards. Also, you can see that the electrolyte has vented through the top of the capacitor in the centre of the image.
Luckily, there were plenty of other machines lying around to harvest parts from!