When you cleaned the pos and neg battery cable connections, did you just clean the ends at the battery? Both ends need to be checked. I have found the pos cable to the starter solenoid completely loose on more than one bike. The negative cable connection to the frame is a problem from time to time.

Your problem just screams that something is not making a connection.

 

You accuratly described the very same problem I had with my 08 Nightster. Try removing the battery, disconnecting the negative cable from the ground connection on the engine case. Reconnect the cable to the negative terminal on the battery, with it pointing straght down, and torque it to factory specs, which is 70in./lbs. Fish the cable back through the frame back to the ground stud. If you do it according to the service manual, it works. By making the connection as you slide the battery back in the holder, it will actually turn the bolt ccw, because it's a very short cable.

 

I tried to "jump start" my Nightster with my truck( engine NOT running) with cables and fried my stator. I learned a valuable lesson. I had to replace my battery, stator and voltage regulator. My mechanic said he had never seen one in that shape before. So my words of advice is NOT to jump it with a car or truck battery.(battery terminal to battery terminal) Too many cranking amps... Take the battery out of the bike or at least pull the Maxi-fuse.

 

I fried my speedo using a battery charger on start mode to jump my bike with a dead battery. Also lesson learned.