1st, do you have the correct solenoid on. an 801 will use a 4 post solenoid. 1 big post from battery, one big post to starter. 1 small post that gets power from the key when it is on, and one small post that gets GROUND from the tranny top mounted starter button.
In other words.. to test your theory that the starter button is bad, set int he seat, and be out of gear. take the little wire off the starter button and touch it to chassis ground. ie, next to the starter button ont he top cover somewhere where you can get ground. if that does not make the solenoid engage.. you likely have the WRONG solenoid.
to test the switch itself, remove that waire and insulate the end for safety. get a test lamp. hook the clip to bat hot. probe the chassis to make sure lamp works, if yes, then probe the lil terminal on the switch, with trans in neutral, hit switch. should make test lamp light up.
all that wsits does is provide ground.
now, if you do have the correct solenoid and your switch is bad then you need to pull the trans cover and repalce the switch. get a new gasket for the trans cover. in a squeeze, you could use some permatex for the gasket.. but it's a booger fix.. don't get carried away.
4spd and 5spd covers are different.. but i'll get to that in a moment.
once cover is off, bend up lock washer tabs, remove nut, pull switch, repalce switch and gasket, add nut and bend locking tabs. notice hole is D shaped so the switch won't spin.
when removing cover, and back edge is a tab that holds a spring and detent ball. it is not under extreme pressure and won't shoot out like a bullet, but the spraing sometimes sticks tot he cover and as you lift it off it then falls into the trans. so once you get the bolts out and get the cover coming up, just go slow and lift the front first.
now. 4spd is easy. put trans in neutral.. center stick.. take bolts out.. lift up, celan repalce, etc.
the 5spd, get to a neutral and i like to be in top 3/r.. but it matters less. remove acorn nunt and bolts. shifter comes off and there IS a key / keyway, then cover comes off.
most of the 5spds fight you... so after the nut comes off.. hit it with penetrant and tap around it a bit.