The bulging component - what does it look like? It could well be a battery or a capacitor, either way you should probably remove it. It will survive fine without the battery, but if it's a capacitor it'll have to be replaced.
Have you tried a second mouse? The Amiga reads the mouse directly, with 2 lines for vertical and 2 lines for horizontal movement. If either of the lines for vertical movement are broken, all you'll get are single-pixel "twitches" when you try to move it. If another mouse does the same thing, chances are it's a trace or connection on the motherboard that's damaged, or one of the chips...
