Ignition cylinder replacement

How are you currently starting the engine and shutting it off? It sounds like the end of the lock cylinder has broken off.

This is what the book says but looking at the parts available I am not sure it is correct, as I recall that is basically the same truck as a Ford Ranger and those use a rod to connect to the ignition switch mounted down on the column.

First disconnect the battery ground cable.
Next remove the screws that hold the lower and upper covers to the steering column and remove the covers.
Unplug the ignition switch connector.
Remove the screws that retain the lock assembly and remove it.
Reverse to install the replacement.

The Ford process (in the picture)starts with removing the battery ground
Then remove the lower steering column cover. Next you turn the key to the run position. As yours is broken you should be able to use the switch rod to move the switch to that same setting.
Next you use an 1/8" punch or stiff wire to push the retaining pin (purple dot and arrow) up in the hole in line with the cylinder under the column.
Then pull the cylinder straight out.
Now look at the green arrowed part. If it is broken off (which I suspect) then you will need to use tweezers or similar to remove the broken piece from inside the actuator inside the cylinder socket.
Then you reverse the process to install the new cylinder, hold the pin in while you insert it into the column and gently align the cylinder to the actuator. Be sure the pin locks in.

For either version:
Test the switch functions and motions before you connect the battery. If they appear correct then reconnect the battery and test again.

One thing you may want to do before you install the new switch would be to have it keyed to match the original keys, that way you do not need to carry extra keys and chance putting the wrong key in and damaging the new lock. Most automotive locksmiths can do that for a reasonable cost.