First question, have you checked for a spark? Take a plug out, put it back in the boot, ground it to a metal part of the engine and crank it. Look for a spark. If none, you have an electrical or computer control problem.
Because the engine will not shut off, and your earlier problem with battery ground, Im pretty sure your problem is electrical, either the ignition switch, or a relay between the ignition swith and starter solenoid.
If you can read an electrical schematic, you can chase this down. If not, take the car to a dealer, or skilled mechanic. You could throw a lot of parts at this and not fix it.