Did you review the color coded wiring diagram I gave you the link to? You will need this to trace/test wires and apply the words of wisdom we are providing.
You will also need to buy a few electrical testing tools to do this yourself, otherwise you will need to bring the car to a shop. You need a 12V light tester and a 12V continuity tester or Ohm meter. To use the continuity or Ohm meter, you want the battery cable off as you do not want 12V going through the wiring.
Well, you could test the starter as well. If you can make a long wire with a small alligator clip on the end and clip the end on the "S" terminal (are you sure you have the purple wire on the "S" terminal of the solenoid and NOT the "R" terminal. If connected to the "R" terminal you will get nothing when you turn the key) and then touch the other end to the battery Positive + terminal, it should energize the solenoid and spin over the starter. If like me, you got the big hands, you may want to drop the starter and either do the same thing on the ground or take it to a shop/auto parts store and let them test it.
You have to eliminate 1 item at a time to narrow down the problem. Begin with the starter/solenoid and then work back from there. With the starter still dropped and the battery disconnected, get a continuity tester - Lowe's has a great one for $29 that has a wireless remote you can remove. Attach one end of the tester to the key/ignition side/wire and attach one end to the purple wire at the starter. (Look at the color wiring diagram, the dark purple wire coming off the solenoid is what you want to test) If the wire is good, the remote unit will go off and make a sound, so you know it is not the purple wire or your neutral safety switch. If it does not, then you have a purple wire issue or neutral safety switch gone bad. It could be the wire itself, the neutral safety switch, or the bulkhead plug where the engine wiring harness connects to the interior/dash harness. You could have corrosion issues or the wire could even be broken somewhere along its path. You can pull the bulkhead plug and use the continuity tester at the starter end to the end at the bulkhead plug. If it works, you are good.
If not, then your wire could be bad/broken inside the casing, or corrosion at the bulkhead. Replace the wire or clean the corrosion. If it works, then do a continuity test from the bulkhead plug where the wire then goes to the neutral safety switch. You will have to locate this, which I suppose is on the automatic shifter and under the console (if manual trans then you don't have this switch). Test it using the continuity tester where the purple wire goes into the switch and then test the switch where the purple wire goes out. Test in both "Park" & "Neutral" shift positions as the switch should work in both positions. Then test continuity from the output side of the switch to the ignition wire at the ignition switch plug. If your continuity tester goes off, it is good, if not, you may have a bad wire - replace or repair and then re-test it to make sure it works.
Install your starter/solenoid and hook up all your wires/battery cable - purple wire to the "S" terminal of the solenoid. With the ignition key plug still removed, find the power wire on that plug that goes onto the ignition switch and test it for 12V with your light tester. You have to have 12V going to the switch which will in turn send power through the switch when you turn the key "On" AND to the "Start" position. If you have power at the plug, you are good. If not, trace the wire like the purple wire and find out where the power begins and stops and fix accordingly.
If you do have 12V, then next check your ignition/key switch. Find the prong on the ignition switch that corresponds with the 12V battery wire coming off the plug. Typically, the back of your switch will have this labelled as "BAT". Attach one end of the continuity tester to this prong. Then attach the other end to the prong that says "SOL" (Solenoid). Turn your key to the "Start" position just like you would be starting your car. The continuity tester should go off IF the key switch is working. If it is not working, then try a few of the other prongs just to make sure the "Start" function is not one of these instead. If your switch is bad it needs to be replaced.
If your starter/solenoid test good, your purple wire is on the "S" terminal of the solenoid, the purple wire is good, the neutral safety switch is good, 12V supply wire to the ignition switch is good, and your ignition switch are all good, then I am lost at this point.