Are you sure the AC power adapter is good here? Given the fact the power cord's LED is not glowing at all I would suspect that first. I wouldn't use a non-Apple unit as there have been fires and people have been hurt by some clone units.
Depending on the damage in your old AC charger it could have shorted power to the signal line. If that happened it could have damaged the comparator logic which controls the charging of the battery. This logic sadly, is on the main logic board. So it may end up needing a new board or getting your system to someone how can replace the SMT components on it.