Def try replacing the battery,
but if the power LED comes on briefly, and no video appears, and the light simply shuts off after a second or two, then it's likely the PAV (Power/Analog/Video) board.
The last PAV board that I tried replacing (about 5 years ago, I think) a lot of the plastics just cracked to pieces - you have to do a fair amount of flexing of the various plastic bits, and age is not a friend of those 2nd-gen iMacs.