If you have tried re-seating the video cables, trying an external monitor, etc, as in the previous posts, and have had no success, then it is still most likely the logic board. If the capacitors are okay there are plenty of other components on a logic board that can fail. If you can remove the logic board, carefully check the soldering points for deterioration or even for the greenish decay left behind by humidity getting into the machine. Especially check out the areas where liquid may have gotten into the machine, ie: the ports, the RAM slots and the optical drive slot.
rab777hp, I did mean the battery connector... Micheal, I'm curious: what was the fix the Apple service center wanted to do? That price is too low for a logic board and waaay too high for a battery connector.
LED #3 is for the connection between the logic board and the LCD. Have you tried reseating the LCD cable? LED #4 has to do with temperature sensors, I think. It's actually good that that one remain off. You say no response from the keyboard: Does CAPS LOCK light up when you press it? Can you insert a cd into the optical drive and then eject it with the EJECT key? If yes, then the keyboard is responding and you are just not seeing it. So, back to no video: Are you able to connect the iMac to another monitor? If the external monitor gets an image, and if the iMac screen is completely dark,then it may just be the inverter that's failed. Can you see a faint image on the iMac screen? Then it may be either the inverter or the LCD panel or both. If the external does not get an image, then it's most likely the logic board. :-(
It could be that the battery cable that connects the battery to the logic board is faulty. Check for corrosion where the cable plugs in to the logic board. This happens a lot, in my experience. In this case, if the corrosion isn't too severe, a toothbrush on the affected area may help. If not, then you may need both the connector and a logic board. :-( Or you could have gotten a defective battery. Or your adapter is faulty. If there is one in your area, Apple has a battery and adapter check that any reputable apple service authorized store/depot will do for free.
I don't know what your level of expertise is, so forgive me if this gets simplistic. Were any of these problems showing up before you changed the hard drive? If not, then the answer is probably with the keyboard - either it is defective or not properly seated. Have you tried using an external keyboard? Also, check the keyboard socket on the logic board for any evidence of breakage, scorching, corrosion or verdigris that is often left behind by liquid damage. If there's damage, then, yes, you've got a logic board problem. :-( Light corrosion maybe removable with a soft bristled toothbrush. When does the kernal panic occur? If consistently on start up, then you may have to simply format and reinstall the OS on your hard drive. Also, make sure that you are using an OS that your Mac can handle. Bad RAM or badly seated RAM can cause kernal panic. Try reseating the RAM, or removing one stick of RAM, or using known good RAM, and see if the problem is resolved. It can be tedious, I know, but if all of the above...