I usually hose them out with Brakleen while the guts are out and the faceplate removed. It's not a bad idea. If there's old, crusty, grease residue inside, solvent ought to loosen it. However, with an assembled speedo, one should be concerned about the possibility of leakage getting inside and causing problems. I've had to open NOS speedometers that had been stored, for decades, face side down...because the factory grease separated then migrated to the faceplate, odometer stack & cover glass.
IDK, maybe if you hold the unit rightside up, then apply a series of short blasts you'll be alright. Bear in mind that it's the area between the square cable socket and outside barrel that needs to be sprayed & lubed...NOT the actual cable end socket, which is solid at the top. That makes the job difficult. Keep white paper towel underneath to catch the flowback. If it's dirty, then the solvent is doing some good, OTOH if it comes back just as clean as it went in, it's a waste of time & material.