The fact that you're hearing the click at the starter even after jumping the slave solenoid means it's most likely an issue with the starter solenoid or the starter itself.
I had a similar issue and I ended up replacing the starter. But before we go that far here is a few things to check.
Take a multimeter and first test the Red/Purple wire lead on the slave solenoid. That should have constant 12.5+ volts. Then test the Yellow/Red lead, that should be zero without the key being turned. Have someone turn the key and then that same Yellow/Red wire lead should have the exact same voltage of the other Red/Purple lead.
If thats the case and the voltages are the same when the key is put into the run position that means the slave solenoid is good.
Next start testing the starter itself. You will notice 3 wires connected to the starter solenoid and one big ground going from the solenoid to the starter body. You want to check the big red battery cable and the small Yellow/Red cable that comes from the previously mentioned slave solenoid. The red should obviously have the same voltage as the battery. The Yellow/Red will receive power when the key is turned. You should have the same voltage you saw at the post on the slave solenoid when someone turns the key.
Now if all this checks out. That means your starter is receiving proper power, and you already cleaned the leads so that's out of the equation.
I was in this same position 2 weeks ago on my 4.3L, everything was working electrically but starter wouldn't turn.
From here I tried just jumping the starter. Using jumper cables directly from the battery i put positive to the big positive post and negative to starter body. Then I used alligator clips to jump the post with the yellow/red wire. (A word of caution tho, I pulled the starter out to do this, be very careful if you're gonna try this with the starter on the engine.) Anyway I got nothing so I knew I had a bad starter and replaced it. It cost me $70 but I had no choice and everything works as it should now.
Hope this helps