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You may look into the device used to flush AC lines. It’s an aluminum canister pressurized by shop air with an extension and a nozzle. You could fill with mineral spirits, fashion an nozzle to aim upward towards the sensors and cogs. Better yet, if you could unscrew the sensors, then squirt directly at the cog with the nozzle. I would drain the old fluid first, save it, and put it back in after the cleaning. Run it for a 100 miles or so, then drain, and refill with new.
Why? Use the old fluid to remove any traces of the mineral spirits that are left behind. You really don’t want to dilute the lubricant over the long run.