You should not have to replace the whole switch, I know on my car b14 the white part on the left side of the switch is where the electrical contacts are, I have had problems with it myself many times and just took it apart and filed the arced part of the contacts down some, you should be able to just replace that rotary switch part, I have gotten 3 of them from the junk yard and rebuilt them and work just like new
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As far as I'm aware the main part of the switch assembly, is 3 separate parts, the lock cylinder, the steering wheel lock, and the rotary switch. If you have ever taken it apart you will be amazed at how easy it to steal a car that way
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All you need to do is take the plastic covers off the column, tilt the steering wheel down, remove the 2 small screws that hold the plastic rotary switch in, and turn the switch with a flat head screwdriver if the car starts when you turn it all the way its the switch, when its all connected together the rotary switch doesn't turn all the way to the stops, and the contacts inside wear out, you can try taking the switch apart (be careful there are parts that fall out very easily, remember how every spring and contact goes) and taking some sand paper to the contacts but that should be temporarily used
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This is for a b14 I'm not sure about b13 but I believe it should be the same
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Unfortunatly I can't take pictures of it all right now because I'm at work but its pretty much straight forward, I have been meaning on doing a how to on this, it works all the same for hondas and mitsubishi cars too