Okay - at this point you are going to need to remove the T/T motor. It is possible that the shaft is broken or has slipped out of its notch where it mates with the hydraulic pump. It is not too difficult to do. From memory...
You need to fully loosen the manual valve and elevate the engine as high as you can get it. In the past, I have backed the back of the boat into a garage, and used a tow strap on a come-a-long. Regardless, you will need a strong overhead beam that can hold the weight of the motor. Turn down the manual lock for additional safety. Remove the large pin at the top of the tilt hydraulic cylinder. Be careful not to bugger up the bushings. That will allow the entire system to tilt forward enough to get to the bolts holding the motor. If I recall correctly, there are four.
At this point, make sure the motor is not loose or cockeyed in its position. If it is, that could be the source of your problem in that it is not properly engaging the pump. Follow the two wires from the motor into the upper cowling and unplug them, and pull the wire to the motor free. It may make it easier to remove the trim angle sensor. If so, take pictures and mark it so you and put it back in the same position. Remove the 4 screws and lift out the motor. Be careful with the large O ring between the motor and the pump. Check the condition of the slot on the end of the pump shaft and the mating slot on the motor.
If the problem is not evident at this time, then make sure that the drive shaft on the pump moves rather freely, and bench test the motor. If the motor needs to be replaced come back to me because there is an API equivalent for under $200 instead of the $500 Honda motor.