Dude! You are the man. Exactly the same thing happened to me. I want to add some detail to this issue: The screw is the culprit. That string of 4 thin wires is wrapped in a protective sheath probably for this very reason, when placing the screen back into place, those wires need to be tucked way down below the edge of the monitor. Otherwise here's what happened to me:
Iplaced the screen in, screwed it down, connected all cables, etc. then when I went to screw the outer bezel back one small screw, as I was screwing it in, went through the protective sheath and actually cut into the wire. This caused a short and the machine shut off. Wouldn't start either. I removed everything and found the wire cut straight through. It was the screw that did it.
So glad you mentioned soldering it, that gave me the courage to do the same, I removed the entire cable from the monitor housing to see if I could free some slack, cut open the fabric shield around the wires, very delicately scraped off the rubber shield on the outside of the broken wire (1 of 4), twisted them together, soldered them, and then squeezed a very small piece of duct tape around them to protect the wires from shorting on another piece of metal.
Made the connection at the sensor on the top of the monitor, then pushed the cable back into place, replaced the monitor, connected the bottom connector (labeled LCD something or other), and then using the plastic tool I got from ifixit pushed the cable down away from all screw ports very gently. Got everything back together and boom, totally works.
When replacing iMac screens, be very careful of that delicate (all of them actually) cable, it's very easy to damage it and like a previous post mentioned, it doesn't ship as a seperate part, so they're not super easy to find. I'm going to buy one today just so I have it in case my repair doesn't last.
Like another poster mentioned, this is the part in case you need to identify it (your model may be different than mine, I have a 24" dual 2.4 from late 2007.
Part: iMac Intel Aluminum 24" LCD Display Temp Sensor Cable .works in: iMac (24-inch Mid 2007)iMac (24-inch Early 2008) 2.4/2.8/3.06GHz.Apple part#922-8236.60
I'll also mention, this is the third drive in this iMac, the Seagate that Apple replaced for me last year died in less than a year. I also saw where the person who replaced the drive reviously nearly cut ther very same cable down by the bottom of the machine near the fan. That means even the apple tech or quite possibly the original assembler of this device had the exact same issue.
Thanks guys, I was so close to giving up.