So after you replaced the solenoid, what happens when you turn the key? If it still clicks (or you hear the starter spinning but not engaging) chances are it's your starter (or the voltages/power getting to it), or at least that's where I'd start looking. My 682 did the same exact thing and it turned out to be a old/bad battery. Replaced it and it's much, much better. I will say, though, that I've heard one too many times that the starter on the KT-17 is a weak point so keep that in mind. I'd try jumping the tractor with another battery or tractor and see if the starter kicks when you do that. If it does, your battery is most likely the culprit. After that, I'd get it running and check the charging system, I believe when running it should be around 13.6v.
Another couple things to check are grounds. So many problems are caused by poor grounds. I'd at the very least remove all your ground connections one by one and clean them very well. Even if that's not it, it could not hurt. As for safety switches, you can always bypass them for testing purposes but mentioning how here seems to be frowned upon I won't go there but it is very easy to do
What I want to know is why do you think you burned up your solenoid?
Edit: I reread your post. If it no longer clicks after the new solenoid, check your safety switches and just for sh!t's and giggles, when you try cranking it next time, make sure the brake peddle is depressed and wiggle the PTO switch, sometimes those can go bad or have a bad contact on them that causes a tractor not to crank. If a safety switch or PTO switch are causing a problem then yes, it would cause a no crank scenario. Either way, clicking of starter usually means it's not getting enough juice to crank the engine. Jump/try a different battery and double and triple check all power and ground connections and switches.