I have moved the question here on the suggestion of the moderator:

I spent most of the day yesterday replacing the 25 1800 uF and 1000 uF capacitors on my mother's old A1058 iMac G5, trying to resurrect it. 6 of the capacitors had obviously bulging capacitors.

Before the replacement, LED 1 would light up, then LED 2 would briefly light and go out. Now LED 2 lights up solid, but LED 3 doesn't light up. I reset the SMU, still can't get LED 3 to light up. LED 3 is supposed to indicate that the screen and the CPU are in communication, if I recall.

I did manage to get a hold of an external display adapter and the iMac is perfectly happy connecting to it, that display works fine! In fact, I have confirmed the iMac G5 does now boot, it just can't seem to talk to the internal screen.

Any suggestions on what could be interfering in the communication between the iMac motherboard and the internal screen and how to test for it? I have been told the LCD displays DON'T generally fail. I imagine it could be the inverter, but don't know how I would test for that.