Bypass the theft system light?

There are ways to bypass the key sensor itself but not the whole system. The rest of the system has to be operational or the computer will not enable the injectors.

Pass-lock sensor bypass:

Look for the black wire and the yellow wire from the pass-lock sensor in the column, cut the wires and you can solder in a 2.2 K ohm resistor connecting them together and then attempt to start the car. It will stall do not turn the ignition off. Let the ignition on for ten minutes or until the theft light goes out. Turn the key off for ten seconds then attempt to start the car. If it now runs you have been successful. If it does not you either made a mistake or there are other problems besides the ignition lock cylinder.

Then just replace the ignition switch and then do a relearn on the system. No programming required.

That car uses the PassLock system with a magnetic sensor inside the key cylinder housing and a magnet on the key cylinder itself to trigger a signal to the security system and activate the fuel system. There is no reliable bypass for it. As for how to repair it, that will depend on what was done to remove the ignition cylinder as there is also an anti-tamper piece in the housing that will damage the sensor depending on how the ignition cylinder was removed. If that was damaged or the housing was damaged so the key cylinder will not stay in properly the only good repair is to replace the steering column to get a good key cylinder housing. Then you can install a key cylinder and do a relearn to make it work. If the column itself isn't damaged it would be possible to replace just the housing.
Can you take some pictures of the area around the key cylinder opening so we can see if the damage is bad enough to warrant a column swap or just a housing replacement?
That car can use the columns from 1997-1999.

If you have minimal damage to the column it should be possible to replace just the key cylinder. For that you have a couple of options, one is to use the VIN number and order two new keys with a lock cylinder from a dealer or automotive locksmith. They will ask for the VIN, your ID and the title to prove you own the car. Then you simply install the lock cylinder and test the new cylinder to be sure it triggers the PassLock sensor. If it does the theft light will go off and the car will run. The other option is to replace the lock cylinder with a new one that comes with new keys, the catch to this option is that the rest of the locks on the car will not work with the new keys. For this method the easy relearn may work, but it may not, some of them are touchy and require a scan tool to reprogram the module. The scan tool would also tell you if the sensor was working or not.
The GM relearn takes 30 minutes, but if the sensor isn't working you won't know it until the process is done and the car still won't start.
The relearn is as follows:

First make sure the battery is good or connect a charger
1-Now turn the key on.
2-Try to start the engine and leave the key in the "on" position.
3-The security light should start flashing.
4-After 10 minutes the light should stop flashing and go off.
5-Turn the key to "off" and wait 5 seconds.

Repeat steps 1-5 two more times for a total of three times.
At the end of the third time you shut off the key, wait the 5 seconds and then try to start the car. If it starts and runs, the relearn is complete. If it still acts the way it is now then either the PassLock sensor is bad and the cylinder housing needs to be replaced as well, or the relearn using that method didn't program correctly.

The key cylinder replacement isn't difficult as long as the housing and column are okay.
As you have a broken cylinder you remove the two covers from the steering column.
Then to install the replacement you need to rotate the key in the housing to the run position with the ignition switch inside the column in that same position. The lock assy. Should then slide in and lock in place. At that point you would do the relearn. If you needed to pull it back apart it is much easier with it already partially apart. If it works you can reattach the covers and should be all set. If the relearn doesn't work you will need a dealer level scan tool to complete the job.

When you cycle the key you are repeating the 2 second fuel pump prime which increases low fuel pressure. It sounds like you have a fuel pump failing or restricted for some reason.

You can't bypass the security system. I have a way to bypass the sensor in the ignition switch but the rest of the system has to remain operational. It doesn't sound like your having a Passlock issue anyway.
Do you have a flashing security light?

Here's how you can bypass the sensor.

Go buy a 2.2k (2200 ohm) ohm resistor at an electronics store. Go to radio shack and just get one as close to that as you can. Remove the covers on the steering column and locate three small wires at the lock cylinder. Cut the yellow wire and the orange/black wire or on some vehicles, it's the yellow and black wires. Solder the resistor between those wires. Tape it up and put the covers back on and follow this relearn procedure. If this doesn't work you have a wiring problem in the steering column.
30 Minute Re-Learn Procedure
1. Turn ON the ignition, with the engine OFF.
2. Attempt to start the engine, then release the key to ON (vehicle will not start).
3. Observe the SECURITY telltale, after approximately 10 minutes the telltale will turn OFF.
4. Turn OFF the ignition, and wait 5 seconds.
5. Repeat steps 1 through 4 two more times for a total of 3 cycles/30 minutes ( the vehicle is now ready to relearn the Passlockà † Sensor Data Code and/or passwords on the next ignition switch transition from OFF to CRANK).
Important
The vehicle learns the Passlockà † Sensor Data Code and/or password on the next ignition switch transition from OFF to CRANK. You must turn the ignition OFF before attempting to start the vehicle.
6. Start the engine (the vehicle has now learned the Passlockà † Sensor Data Code and/or password).
7. With a scan tool, clear any DTCs if desired (history DTCs will self clear after 100 ignition cycles).

That would indicate there is another problem.
You really should start a new question to get some help.