You can complete a mirror glass repair by yourself by removing the broken glass and replacing it with new pre-cut glass. Many manufacturers sell pre-cut mirror glasses, according to FamilyHandyman.com
To repair a broken mirror glass, first remove the original glass remnants and clean the base with glass cleaner. Visit an online or local store to order a pre-cut mirror glass of the appropriate size and shape. Although most new glasses come with adhesive on the back, you may add some sealant both on the surface and on the back of the glass for extra-secure adhesion. Place the new glass on the base and press lightly. Hold the mirror in place with masking tape until the sealant cures.