Well, if you hit a drop out of order it makes a sound and resets, right? Except the king is down and can't reset, so the game just keeps playing the sound and trying to reset it until you progress to the queen at which point the king is no longer out of order. Then when you hit the ace and it resets the whole bank you are back to the same problem.

Glad to hear that was it! You said the King target had a burnt coil, so that has been replaced but still doesn't fire? If so your first stop will be the associated driver and predriver transistors on the driver board.