to see if it's just the solenoid, fit a charged battery, or jumper leads to the battery leads, and try starting it.
you may need to hold the front brake lever on for it to fire too.
if you still dont get anything, try and jump the 2x smaller terminals on the starter solenoid, or run a seperate power to the starter motor side of the solenoid.
the electric start not working, could be the terminals inside the stater solenoid are corroded.
if you take the rubber cover off it, you'll see a couple of tabs holding the top half to the bottom.
use a screwdriver to pry them up enough to seperate the 2x halves and it'll come apart.
take note of what sits where too.
Yeah $50 each is ok, you'd have that much in parts etc