How can I fix a scratch on a mirror?
If you can feel the scratch with your finger nail then there is a good chance that it is too deep to remove by yourself, but you can certainly try!  Removing scratches from glass is similar to removing them from car paint - you use a polishing compound (made for glass) and ideally an orbital polisher to try and remove the scratch (lots of muscle power works too).  Make sure the area is very clean before you start and that all polishing pads, cloths, etc. are also very clean.  If there is any dirt then when you start polishing, you will end up scratching the glass even more by grinding the dirt into the surface.