Cleaned battery corrosion, but A and B buttons don't work

Hi! I came upon my brother's old original Gameboy yesterday. It had battery corrosion from years ago; I don't know much at all about electronics but have fixed a few toys with battery corrosion before. Gameboy wouldn't turn on; I took it apart and cleaned all the corrosion I could, first a little with baking soda/water and then with vinegar....Turns on now and everything works, EXCEPT for the A and B buttons, obviously making gameplay impossible...Links to pics below: is there anything I can do to get it working/what part is keeping it from working? I realize it's not completely clean but don't want to spend more time on if I've broken something beyond repair (some of the green...plastic? started "melting" off on the back side....was vinegar not supposed to touch that....?) If it can't be fixed, I guess we'll at least have a very cool Super Mario World 1-1 music player, haha. Thanks!

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