I lost the key to my yamaha blaster so I just wired a toggle switch to it and ran it like that for about a week till I found my ignition key under a couch cushion. Also from the sound of your problem you either have a faulty kill switch like 06F150STX or the spark is too weak to make the gas ignite. Although the plug may be showing a little spark it can sometimes not be enough. If you haven't already go to the parts store and buy a new plug. If that doesn't work then I would get an ohm meter and test all the coils and see if their producing the correct amount of voltage. Also check every wire and electrical connection on that atv something may be loose or came unplugged. A wire may be naked and have a little bit of moisture creating a misfire. If all that turns out good then backtrack yourself to the carburetor. Make sure that you put all of the jets back in correctly. Even the slightest change in the pilot jet will prevent it from cranking. That's the jet that controls the fuel/air mixture at idle. Their should be a fuel/air mixure screw on the side of the carburetor somewhere,not the idle adjustment screw. This screw is usually small and the screw head is flush with the side of the carb it allows for a fine tune adjusting of the fuel/air mixture at idle. If it's not see correctly then the atv will not want to start also. Make sure that the needle jet doesn't having any trash blocking it's path. Their could be a microscopic peice of dirt in their that is prevent the float from closing all the way and dumping too much fuel into the motor. I know you said that the carb has already been cleaned but I have had trash get into my carburetors before and it would take a good purge of air from an air compressor to blow it out. Try to find a website with a diagram showing all the factory fuel adjustment settings and voltage requirements for all of the coils. From the sound of it the atv your stepson has is one of those chinese brand knock off's. Those things are stupidly unreliable and are like a tampon. They should be called the disposable atv. They are cheap cause they are made cheap. But are a blast to ride when running properly. Hope this info helps you out.

 

One of the valves was stuck and the other way to loose . Was going to try the toggle switch but it has 6 wires on the ignition in 3 different spots . I think the red and black are the ones I need , but not sure

 

Yea that may be your problem. I adjust the valves on my atv's every time I change the oil which is about 10 hours of riding time.