Do you have a voltage meter? I know it sounds elementary, but if the starter solenoid is not getting power, then trace the wire back to the source and make sure the connection is in tact and that there is power where it is connected. It sounds like everything else is working properly. Are you sure you got a good solenoid?
The only other thing that would make the click you're hearing if it's not the starter solenoid or the starter itself, is a small relay engaging. That's probably on or near the control circuit board. It is probably switching power to send to the solenoid. The solenoid acts as a switch bewteen the main power source (battery) and the starter. There should be a signal wire from the circuit board relay that causes the solenoid to energize and switch the main power to the starter. Test your signal voltage to make sure it's right, and test that you actually have 12VDC on the supply side of the solenoid. And again, it may sound elementary, but make sure the solenoid has a solid ground connection....that's usually obtained through the mounting screw. A solenoid is just a big, heavy-duty relay that temporarily switches the main power from the battery to the starter.