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If you have remaining length in the end of your LED strip, clip it with scissors in a location that will allow you to attach the wires from your power source while still being hidden behind the mirror. Attach the wires from your LED strip to the wires for your plug with crimp-on wire connectors, which are available at most hardware stores. You have the option of soldering your wires together instead of using wire clamps if you are experienced with soldering and have the proper equipment. Remember that soldering will be permanent, as opposed to wire clamps which you can remove if you ever want to switch out your LED strip for a different type of light. When your lights are attached and connected to a plug, you can hang your mirror so that the plug reaches your nearest outlet. Rehang a previously used mirror using the hardware that you used on it before, or hang a new mirror using the hardware that it came with. You're all ready to plug your mirror in and enjoy the backlight effect you created. If you want to be able to turn your lights on and off with the rest of the lights in your room, you'll need to speak with your electrician. They can instruct you in drilling a hole in the wall behind your mirror to run a transformer into the existing circuits in the room, or they may offer to do the job for you. Tell your electrician that you've created a backlit mirror and you'd like to connect it with rest of your circuits for whatever room that it's in.
Connect your lights to your power source cable. Hang your mirror close to an outlet to manually plug it in. Consult an electrician to control your lights from a wall switch.