To clean, just spray CRC CO Contact Cleaner everywhere into the switch, including through the key slot. This stuff is not oil based, and wash dirt and oxidants out, leaving the switch dry. Soak it in one or two long sprays rather than short burst. If it does not come right, you can always opt for the switch driving a relay until you find/get a new switch. You should be able to get one that will handle a decent current while drawing less than 0.2A, which at 5 Ohm is only a 1V drop over the switch (i.e. the relay will see about 11V).

Your wiring diagram is for the early model, the switch is for the late model. My diagram is not very clear (as they usually are), but I will e-mail it to you.