The stuff that leaks out of the Varta is potassium hydroxide. (KOH) Not an acid, but a strong base. It needs to be neutralized with an acid. Some of the deposits of KOH very quickly form a cap over themselves of potassium carbonate from CO2 in the air. Vinegar is not strong enough to remove the cap. If it is not removed, it will go on corroding copper and tin underneath, (traces, IC pins, and solder) until some necessary signal trace or IC pin connection gets eaten and the board dies.
Vinegar does help early on tho. Let it sit and brush it very lightly with a soft brush. Both sides of the board near the Varta. (Of course the Varta has been removed, right?) Remember the KOH is very corrosive to skin and especially to eyes! There may still be some wet, active stuff under the deposits. If it foams, it means you found still active KOH. Chances are that most of it has already reacted with copper or tin tho...
Then wash the whole board in plenty of warm running tap water. Try to flush out anything from under the chips. It will remove most of the water soluble potassium salts you created with the vinegar. Then rinse the board WELL with distilled water. You can put some isopropanol in the final rinse if you want to make it dry faster, but none of the salts are very soluble in straight alcohol anyway. Shake the loose water off or blow it off with air. It needs to dry for at least 8 hours, or you can speed it up with a hair dryer on low heat. If you put it back in the machine still wet, it probably will not work at all. Water gets trapped under the IC's and in the IC sockets. Don't get impatient.
If it was very badly corroded, parts may have literally fallen off the board by this point. This would mean a vary hard to refurbish board, or just game over.
You will probably see some whitish, or blue-green 'corrosion' left usually around IC pins. Those salts are not water soluble and can only be removed with a chemical I will not name. (Dangerous). In most cases the white deposits are tin salts, and do not conduct electricity. The blue-green ones are copper salts, and only conduct if moist. If the solder connections to IC's etc are now mostly tin and copper salts, and not conducting well, major work is needed to save the board. If it's even reasonably possible.