Car will stop running just after it starts

You could have damaged the pass lock sensor when you drilled it also there set up to be turned with the key not a screwdriver. When I went to Saturn L series training I was told it will also defend against people doing that and cause a no start condition. You can try replacing the key cylinder blow all the debris out and clean it well before installing the new ignition cylinder. If that doesn't work then we can try relearning the pass lock sensor. If not then the ignition housing. If you brought the car to me I would just suggest you replace the ignition housing and cylinder since you drilled it. If I would have got to the car before you messed with it I probably could have freed up the cylinder removed it keyed the new cylinder to your old key and installed the new cylinder.

You might have even damaged the sensor on the ignition cylinder.