You will need:
Round Mirror (best place to find one of these is inside your make up compact case)
Orange Power Glue remover (or similar product which dissolves glue)
Poster Board or craft cardboard
Fast setting glue.
Start by placing a few drops of glue dissolver around the edges of the mirror. Using a knife or box cutter very gently slide the blade between the mirror and the compact so that the glue dissolver and run under the mirror to reach the glue. Keep adding drops of glue so that it can run underneath the mirror. Be patient and give the glue dissolver time to work.
Once you have the mirror free from the compact you can measure a length of cardboard to use as your mirror hanger.
Use fast setting glue to attach the length of cardboard to the edge of the mirror - in effect the cardboard is just for show. Always attach the mirror directly to the dollhouse as it is heavy and should be secure from children prying it off.