I think I found the issue.

When I was pressing on the coil, I managed to get the LED to briefly go off a couple of times. I unsoldered the coil and checked with a multimeter and found that the circuit wasn't complete. I then checked around the area that had been hit with the balls a few times and found the break to be there. As the coil is really small I tried dabbing a bit of solder where the break was hoping that this would solve the problem.

Unfortunately, whilst the coil is now continuous, the LED remains lit. I assume that the solder fixed the break, it may have changed the resistance of the coil, and the ciruit still thinks the ball is present. Oh well, it looks like I'll be looking for a new coil (A-16393) which doesn't look promising. If I can't find a coil, it looks like a replacement board set.