Makes no noise at all when turning the key?

Our son left his car lights on all day at school (1988 Honda Accord). When my wife and son first tried to jump-start it, they crossed the cables (positive to negative). Now, the car won’t start. It makes no noise at all when turning the key. We tried it in park and in neutral. We tried jump-starting with new cables. I checked the water in the battery and it was slightly low so I refilled it. We hooked it up to a battery charger and the charger says the battery is fully charged. The interior light comes on when the door is opened, and if the key is turned in the ignition and the door is open the warning signal sounds (ding ding ding). However, with the key turned in the ignition, none of the other electrical components (e.G. Fan, radio, etc.) Work. There is one 10-amp fuse that is blown. But, when I moved one of the other 10-amp fuses to where the blown one was it made no difference.
A couple weeks ago we were having starting issues because our son lost the gas cap, and didn't replace it for 2 weeks so the fuel filter got clogged. The repetitive attempts at starting killed the starter motor so we replaced that too. The neutral starting (safety?)Switch is a little finicky. We occasionally have to take it out of park and put back in to start it, but that isn't helping now.
Can you suggest where to start before we have it towed to a shop? I am thinking about a new battery even though this one supposedly has a full charge.