Breaking an antique hand held mirror can be distressing for a lot of reasons: sentimental value, devaluation of the mirror's worth and the inconvenience of having it replaced. It needn't be difficult to find a replacement glass for your hand held mirror. Cost wise, you'll have to decide if you want to replace the mirror with a flat edge or order a custom-cut beveled glass replacement. Either way, help is as near as your local glass shop.
Call your local glass shop and ask if they handle broken antique glass in hand held mirrors. They can usually replace the glass itself if it is flat and does not require a beveled edge.
Send your mirror to an antique glass specialist if you decide you want to keep the original mirror's beveled contours in tact. Your local glazier will have a reference if they do not do the work themselves and may offer to send it for you.
Ship your mirror if it needs a glazier not in your area. Cut a piece of cardboard a few inches larger than size of the mirror. Put a dab of school glue on the back of each shard, and place them in order on the cardboard. Allow to dry. Wrap cling wrap around the glued mirror and cardboard horizontally and vertically to make sure the pieces do not fall out if they become dislodged from the cardboard. Place another section of cardboard over this and secure it tightly with rubber bands. Wrap cardboard in enough bubble wrap to keep it from moving inside the box and ship as usual.
Be sure to insure your package with the exact cost of the entire mirror and not just the cost of replacement.