There are a bunch of theories on this. I've run up against it a bunch in pinball and arcade, one of my other hobbies. Some say to use vinegar while others recommend alternatives. I've used vinegar to neutralize the corrosion and a combination of toothbrushes and Q-Tips/cotton swabs. Then I rinse the area with distilled water, then clean it again with isopropyl alcohol (which will help remove any leftover water). You can speed drying times with compressed air (and blow liquids out from under the chips). I'd strongly recommend letting the board sit for a few days before trying to reapply power to it after cleaning.
First thing first, GET THAT BATTERY OFF THE BOARD!! Just make note of your BIOS settings and use a diagonal cutter to snip it off the board. Clean up the area as above, then desolder the pins and connect your replacement battery. For the replacement, there are a ton of options. I'd strongly recommend something that you can mount off-board. Some of the pinball guys put their battery packs way off-board in a little Ziploc bag in case it leaks again. Either way, I'd recommend a non-rechargeable replacement. Depending on the board you can find varying solutions. My preferred method is to use a 4 AA holder with a diode in place of the fourth battery (in line so that it prevents charging the AA batteries). In this scenario, that will put you slightly over the required 3.6v so I'll defer to others regarding if that's OK.
Another option I've seen is to use a lithium battery in a holder, something around a 2025 or 2032 would do well. I have used this method, and upon inspecting the board sometimes you can find some creative ways to mount the holder. I've noticed that most boards have very little in the way of traces/layers running below the battery so you can get away with drilling a small hole for one of the contacts from the battery holder. Use the factory contacts for the other lead. You can then traverse the space from the drilled hole to the factory contact with the charge-blocking diode.
This supplies around 3v which should be sufficient.
Edit: Forgot to mention, this corrosion doesn't look terrible to me. I'd also like to point out that the corrosion will do a number on the solder pads, I've had some luck restoring them with a few liberal applications and removal of fresh solder. I haven't tried it yet, but I've also considered trying to use DeOxit for this purpose. It's expensive, but I've heard it works wonders in other applications (not sure if it works with battery corrosion).
