If your posts are pitted, you can just directly solder a wire between each pin and post as shown in the first photo. This may not be necessary if your connection is restored simply by removing the puffy battery (button will illuminate when 16-24VAC is applied to backside screws and unit held together).

Taking out the puffed battery may be sufficient to let the pin-to-post contacts close, that didn't work on my unit, so I soldered wires from post to matching pin, although polarity doesn't matter, since it's the 24VAC input supply.

Use caution soldering the pegs pressed into the back housing so you don't melt the plastic or change the angle or position of the brass pegs so the unit won't close fully.