Shaving foam
While shaving foam might not be the first item you’d think to use when tackling a cloudy mirror, it can be very effective. Squeeze a thin layer onto your finger and wipe it over the mirror, or apply it directly to the surface. Be careful not to use too much foam, as doing so will leave streaks.
Then, take a clean, dry cloth and wipe the foam away until you can see your reflection. The results should be effective for several weeks. When you start to notice the mirror steaming up again, repeat these steps.