Besides the ignition switch, there is either the clutch switch (manual tranny) or shifter switch (auto) that allows voltage to the starter soleniod control circuit. If that is bad, the soleniod won't close even if the ignition switch is good.
Also check fuse 29 (60A) under the hood and fuse 34 (10A) below the instrument panel.
Pull the wire off the small terminal of the solenoid, and check to see if you get 12V there when you turn the key with the clutch pedal to the floor or shifter in park or neutral. If you do, then ignition and clutch/shifter switches are OK.
When you jumper the soleniod, are you jumping across the 2 large terminals or applying 12V to the small terminal? If the latter, your soleniod is OK and the problem is indeed upstream.
By the way, the ignition cylinder is separate from the ignition SWITCH, which is under the instrument panel and connected by a pushrod from the cylinder down thru the steering column. So changing the cylinder doesn't help with an electrical problem.
If you can't track it down, post back and I can get you the troubleshooting for the igntion switch (or you could just replace it).