This is one of those problems that you might need to use a "divide-and-conquer" approach to resolve.
Divide the problem by determining whether its the fuel sending unit or the wiring and indicator light. You'd have to have a diagram of the circuit to do this properly.
But it would be something along the line of determining if the light stays on all the time even with the fuel sending unit disconnected. What I'm saying is that the fuel sending unit may be OK, but the wiring shorted somehow.
Hope this helps.