Cleaning vinegar's 6% acidity works well for this task. Try Amazon's Harris Cleaning Vinegar, which also leaves a pleasant orange scent. For baking soda, we recommend a pack of two boxes from Walmart.

Harris recommends always cleaning with the grain to remove more ... you can brighten it up by lightly spritzing ¼ cup of white vinegar and one cup of water onto the mirror, then wiping it away ...