The most likely problem to the starter button not activating the starter is one of the safety circuits is being engaged and preventing it. That is going to have to be systematically tested and repaired. At this point, it is highly unlikely it is the actual solenoid has failed. Not impossible, but improbable. Improbable enough that I would not spend money buying a new one just to test the system.
If the neutral light is not on and the clutch switch is not working, the starter won't work. The easiest way to test that is to remove the side stand relay from the system and re=test. That relay should be under the seat and have a blue/yellow wire, a red/white wire and a black/white wire coming out of it. Find that and just unplug it and try the starter. If the bike cranks then either the neutral switch has failed, the clutch switch has or the relay. You can check the clutch switch for continuity and the same for the neutral switch. The clutch switch is just wire to wire, and the neutral switch (with the transmission in neutral) from the center of the switch to ground. If those two switches check out ok, then the side stand relay has failed.
There is another remote possibility that the starter button contacts are not making contact too. Again, a quick continuity check can reveal that.
As far as the box of rocks, there is no quick easy solution without know what is wrong. There are a number of symptoms that produce the same sounds. State of tune, IE valve adjustment, carb sync, idle speed, idle mixture can all affect that. The easier the bike starts, the less obvious that problem is. Also, depending on how long it was abused by being started with other problems, it could damage various gears in the starter system too. You need to get the bike running first before you can sort some of those things out. The valve adjustment can be done before it starts, which it should be since the engine has to be cold but all carb tuning requires the bike to be running to make final adjustments.
I saw your other post on the 2 batteries, and my question is why? What are you trying to accomplish by using 2 batteries and why do you think you need them? as far as connecting them, if you connect them in series you will get 24V at the same amp hours. If you connect them in parallel you will get 12v @ double the amphour rating. If you try them in series, you are likely going to start burning up the 12v components on your bike.
If you have any more questions, feel free to ask.