Most older Miami Carey Mirror frames permit disassembly by removing the screw at the inside bottom of the frame. The steel frame must be carefully spread apart. Then the mirror, cardboard spacer, and steel backing sheet may be removed. Before doing this, I usually remove the mirror assembly from the frame of the medicine cabinet to prevent damaging the hinge or frame. The hinge is usually riveted and will require a pop-rivet gun to reassemble. If you are not comfortable replacing the mirror take it to any commercial glass shop.
Replacing the silver backing is rarely performed these days unless the glass is a priceless antique. VERY few shops can provide quality silvering and those that do are quite expensive.