The pilot's improper decision to attempt a single-engine missed approach with the airplane in a slow airspeed, full flap configuration, which resulted in a minimum control speed (Vmc) roll. Contributing factors included an inoperative engine for undetermined reasons, the pilot's in-flight decision to divert to an airport with low ceilings and visibility while better conditions existed elsewhere, the pilot's failure to advise or seek assistance from air traffic control or his company, and the low cloud ceilings, fog, and night lighting conditions.