Yeah, with situations like these it's best to start bare-bones, taking all unnecessary cards out. Make sure everything is seated properly after a thorough examination of the board and components (especially capacitors). Take a look at the memory and see what kind its using - 30pin, 72pin, single sided, double sided, 60ns, 70ns, ECC, Non-ECC? Try swapping the memory if you have some laying around.
Is the video onboard? If so, maybe you could try throwing a discrete video card in there (ISA, or VLB if the board supports it), and see if you can get video that way. Here again, a jumper may be needed to enable/disable the onboard video controller if it has one.
I usually replace the older AT power supplies with a newer ATX one, using an ATX to AT converter cable. The last thing I want is an exploding PSU or a situation where the PSU destroys something else.
As far as the battery is concerned, I use these battery packs in at least 3 or 4 system and they've worked just fine. I've just used Velcro strips to stick the battery somewhere in the case - within reach, but out of the way of wires and other components. The jumper to enable it is most likely one of the ones real close to that battery header (or there is the off-chance that it may just... work!).
You've already taken care of that battery which is absolutely vital. I don't see too much corrosion there, but if you want you can take some vinegar and rub a little on areas where you see corrosion. This should neutralize the acid, and a toothbrush does a good job of getting the acid corrosion 'cruddies' off the board.
This looks similar to your board layout (at least from your pics):
http://museum.ttrk.ee/th99/m/M-O/32131.htm
My Retro B:\ytes YouTube Channel & Retro Collection
