Powers up, no chime, three LEDs, then shuts off.

I found a few posts with similar issues here but nothing exactly matching my situation. Of course, this is my personal machine. :-)

My iMac is generally running 24/7/365. Two days ago when I got home I discovered my house had suffered a power cut; the microwave clock had reset itself and my iMac was off. I was unable to turn it back on. I unplugged, waited, then replugged and pushed the button. I could hear the fans kick on and the optical drive do its little POST, then it shut off again. Once that happened pushing the power button did nothing; I had to unplug/wait/replug to try it again.

I have it on the bench now. LED 1 lights up when it's plugged in. LEDs 2 and 3 light up when I push the power button, but ~3 seconds later LEDs 2 and 3 turn off followed immediately by LED 1. LED 4 never lights up.

Unplugging the screen's power and signal doesn't change anything.

Memory tests good.

Resetting the SMC does nothing. Resetting the PRAM is impossible.

According to this guide, no LED 4 means the computer and display panel aren't communicating. BUT! There is one capacitor on my power supply which appears to be ever-oh-so-slightly leaking (in the middle):

Prior to the power cut the iMac had zero problems. I'm thinking my plan of attack would be to replace the capacitor, see if that brings the screen up, and if it doesn't then just replace the screen. Good idea? Or should I just bite the bullet and replace the screen right away? Or is there something else I should check first?

EDIT: I can't find a pinout or schematic for this power supply, but I measured the voltage anyway using this video as a starting point, even though it's for a different iMac model. Facing the connector on the board with the retaining clip on top, the upper-left conductor is Pin 1 and the lower-right is Pin 12.

Pin 1 = 0

Pin 2 = 0

Pin 3 = 0.09

Pin 4 = 0.09

Pin 5 = 0.09

Pin 6 = 0

Pin 7 = GROUND

Pin 8 = 0

Pin 9 = 4.21

Pin 10 = 0

Pin 11 = 12.76

Pin 12 = 0

According to the video I should be getting 12V out of Pin 11 but since it's a different iMac model I'm not 100% sure.