2008 Jeep WK Alternator wire damage won't start just click one time

Unless you can verify those burned wires are from the alternator, I think something shorted and drained the battery or the damage is preventing starter operation. Under most configurations, the alternator has two or three wires; one fusible link wire cable connected to a terminal on the alternator (hot all the time) and a separate wire going to another terminal for the battery light. A third wire on some vehicles is a control from the ecm to turn on the alternator while all others are self regulating. Without more info, you may need expertise from a repair shop capable of determining what caused this short before repairing the damage.