Broken underground power line and dead trees

RE: Could the discharge of electric current from the power line and the combination of wet ground have killed the trees?
Yes, it is possible. If you have two conductors and salty soil between them there is electricity leakage that can fry everything ... not only trees. If there is some leakage there must be some evidences for it, on the electric meter of the switch-board (if it has fiscal memory).
But this may not be the only reason. If the rain has been acid (too much CO2, NO2 and SO2 dissolved in the water) your trees would have been dried ... with or without the electric cable.
The dying trees are very serious indication that we are the next ones, for we breathe the very same air that has caused the acid rain.
You can investigate the case - see whether the cause of the death is from the roots (electric underground current) or from the branches (the acid rain causing defoliation). You can also take samples of the soil within the roots and see what is its conductivity. Besides electricians you may consult also a botanist or a phytobiologist, and an environmentalist.