The ignition wire problem is totally false. The way a solenoid works is when you send it twelve volts, via your starter button, it closes a circuit which gives continuity to the two posts (one is connected to the positive side of the battery and the other is connected to the cable that runs to the starter. The clicking is the sound it makes when it closes. This means that your problem is one of two or three things.
Either your starter is bad so that the solenoid is sending 12v but it is not doing anything or your solenoid is bad so it is clicking shut but isnt actually closing the circuit so n o power is going to the starter motor. The other problem could be that your battery is dead.
This is a very easy diagnosis. All you need to do is hook up jumper cables to a car battery or your atv battery. I prefer to use a different battery so that i can rule out a dead battery as an issue. So first hook up the jumper cables to a battery that you know as good such as one in your car. Now connect the positive side of the jumper cable to the post that is on the starter where the cable is bolted to. now touch the negative side of the jumper cable to the case of the starter a couple times in a couple different places. If the motor turns you have ruled out the starter. If it doesnt turn buy a new starter. If it did turn know hook the jumper cables up to the battery on your atv and do the same thing you just did. If it doesnt turn know your battery is dead and either needs charged or replaced. If it does turn than you know your solenoid is probably bad. But before replacing it go through and disconnect both of the battery terminals as well as the two big cables on the solenoid and the one at the starter. Disconnect them one at a time and sand the cable and where it screws on. If it still doesnt work it is the solenoid. Another thing you can do is touch one positive side of the jumper cables to the battery side of the solenoid and the other positive side to the starter post of the solenoid. If it turns that is verification that your solenoid is bad.
Starting systems require a lot of amperage which is why they like to fail. It is about one of the easiest circuits to diagnose on a vehicle just take your time and go though trying to find where the amperage is making it to. Simply using a multimeter to test voltage doesnt tell you much since more often than not you will be getting 12v it is just not nearly enough amps so the motor wont turn. you can use a mulitmeter and check things first but all it will do is tell you if some is bad it wont tell you if it is good. Ie if you have 10v at the battery than the battery is bad but if you have 12v it may not necessarily be good. Same thing with the solenoid. If it doesnt have 12v on either post when it clicks shut than it is definately bad but if it does that doesnt necessarily mean it is good.