First of all check the transistors that drive the coils.. these are Q1 through to Q24 on the bottom right of the driver board.
Transistors are a BD649 - Jaycar carry these or an alternative.
If the transistor is burnt out, it should be obvious to spot... once burnt out like you had they tend to fry. The resistor next to it will more than likely be fried also these are a 390 Ohm 1/4 W but its better to use a 1/2W.
Not all the transistors are used and its possible to pinch a spare and swap.. but youll need game specific diagram to find out which is which.

If replacing the transistor doesnt work, then it will also be IC 17 through to IC20 - One of these will be associated to your transistor... once again.. youll need game specific diagrams to ascertain exactly which. Or you can test them yourself
The IC's are 7407 buffer chips... real easy to test and sold everywhere... make sure you use a straight 7407 .. not the HC or LS version... these are an input in and then straight output... so what goes in then comes out..