What you describe I have seen several times on Briggs, Kohler, and Kawasaki engines. It seems after the engine gets wet, either from washing, or high humidity the modules draw moisture and will cause the symptoms you describe. And normally I find it will effect both modules not just one. I would also bet that if you inspected the epoxy on the modules will have cracks in it, especially around the kill wire terminal. Most likely you will need to replace both modules to repair the mower.
I would also bet if you disconnected on spark plug, and it doesn't matter which one, and try to start the mower it won't start.