If you have a completely dead battery draining the system you probably have less than 12 volts even when plugged in to shore power. With less than 12 volts many 12 volt systems will not operate. You have some warm water from the 120 volt element. Disconnect the battery and try to light it with out it connected to the battery. Many convertors will operate all 12 volt systems when plugged in without a battery connected. With a dead battery connected, even while it is trickle charging your pulling your volts well bellow 12 volts.