Key will not turn to start the car

The cable should not be the problem. It is designed to prevent you from turning the key off while the car is in gear. It allows you to turn the key to the lock position once the shifter is in park. The most common reason the new cylinders will not turn is that many of the replacement units are made to poor tolerances. This is easy to tell, if you can wiggle the key and it now turns, the key or tumblers inside the cylinder are worn already.
I have seen where a bit of graphite lock lube can help. You might talk to a locksmith, it might be possible to cut a better key to make the cylinder move better.

I have seen some of the after market units fail when installed. To test it you will need to remove it and see if it is the cylinder or the housing. It is possible that the ignition switch itself has failed internally so pulling the cylinder itself would let you test that as well.

Yeah, the only way to get the cylinder out in that case is to drill through the side.
Here is the book method of replacement in that event:

Unbolt the switch from the dash and lower it to gain access.

Use the following procedure if the ignition switch lock cylinder is seized or will not rotate:

12.1. Protect the immediate work area with suitable material such as clean shop towels or a clean fender cover.
12.2. Locate the surface for the release button (2) for the ignition switch lock cylinder on the plastic ignition switch housing. Center punch a location (1) on the rib approximately 3/8 inch reward, toward the key entry end, from the cylinder release button.
SEE ATTACHED IMAGE BELOW

12.3. Carefully drill a pilot hole through the plastic housing with a 1/8-inch drill bit.
12.4. Using a 9/32-inch drill bit, carefully drill a larger hole at the pilot location, and slightly into the surface of the ignition switch lock cylinder, to break the release button retaining spring.
12.5. Remove portions of the broken spring from the hole using a small suitable tool.
12.6. Grasp the ignition switch lock cylinder and remove the cylinder from the switch housing.
12.7. Remove any plastic flash from the drilling operation, and using compressed air blow out the ignition switch assembly.

INSTALLATION PROCEDURE

If the lock cylinder would not rotate or was seized, perform the following procedure:

1.1. Install the new ignition switch lock cylinder as required by rotating both the cylinder and the ignition switch to the ON position.
1.2. Push the ignition switch lock cylinder into the ignition switch housing until fully seated.
1.3. You may have to slightly depress the release button as the button passes by the 9/32-in hole previously drilled in the ignition switch housing.

Where did you get the replacement unit from?

Good luck. Let us know what happens.