Nice score! A bit hard to tell,but are the traces damaged around C113 where the battery was? kind of looks like it in the picture(this needs fixing first if so).I see green corrosion on the resistor in that are as well as a IC.If its not damaged first thing i recommend is check voltages (+5 and +12) with the psu hooked to the board and power it up with the 3640 attached.if voltages are ok, shut it down and remove and reinstall every chip,look for any corrosion on pins or socket. Don't just push on this chips,it does no good, and use a proper plcc puller,the sockets are delicate-DONT TAKE SHORTCUTS! The missing key switch stuff wont matter, it just had a reset in it when you turned the key full and a kbd lock,it will run fine without that.
Carefully pull each zip ram chip and reinstall,again looking for any corrosion on the pins etc. half the chip ram zips is usuall socketed and all the fast ram zips.
you can use vga on the 15pin connector out the back with the switch on,if no video there i would hook a 1084s up or some such to the 23 pin port. hopefully it powers up by now. If not check the 3640 well,capacitors leaking can cause a eaten trace on it as well,sometimes under one of the capacitors.
Hopefully the above gives you something to work from.
Mech