I can tell you a little. If you cant get the range lever far enough forward to actually get the transmission in HI, there appears to be a set screw on the lever, where it goes on the shaft. As far as indicator light, I can tell you how it works, but you won't believe it.
The range lever has a bar that goes over to a lever that acts as a bellcrank on a 2000 ohm potentiometer. With 5 volt applied on one end of the pot and ground on the other, the varying voltage on the center (wiper) of the pot is fed back to the believe it or not the cruise control module. An integrator in the cruise module then detects the varying voltage and decides which gear the transmission is based upon this voltage, and sends the appropriate signal to the display module to light the proper L N M N H lamp.
But all of that aside, you are saying that you can't get the transmission in HI range, so that is a purely mechanical problem, and I would look at that shaft and the range selector lever, and what looks like a setscrew to me. Again, the indicator lamp has nothing to do with what actual gear of the three the transmission is in, it is just for your viewing pleasure.

Just like dialtone has nothing to do with dialing numbers on a telephone, it is just to make you happy.