hpst I'm the same as you where I have not found one LED-backlit computer monitor that is proven tolerable for my own use for an extended period.
We have a Mac station at work to do Apple App Store iOS promotions on, and apparently our Desktop Support team has mentioned there are no more CCFL-backlit monitors left in inventory (like the Dell U2410 that I'm using right now), and that all they have left are LED-backlit monitors that all give me instant problems like the Dell P1913.
Last month though, I faced an unavoidable situation where I had to use that monitor for about 30 minutes. About 4 years ago, 30 minutes of exposure to one of these displays once gave me a splitting 4-day migraine that was horrific. That was while I worked together with someone at their desk, where their settings would be default. In this situation, I cranked the brightness to 0, changed the color temperature to Warm on the OSD, and wore a pair of polarized sunglasses. I was pretty amazed that after 30 minutes spent at the end of a work day, I had no issues after at best, and mild issues at worst. It didn't cause a lingering migraine.
However, looking at the display for under 1 minute a few days prior to this caused a 3-4 hour migraine. That situation may have been default settings w/ no sunglasses.
Curious, have you tried sunglasses (polarized) against an LED before?