There is a specialty lighting store near me that I am going to soon with photos to work on lighting at my house. I will show them your photo and see if they have an idea. Probably won't be till Fri or Sat but will get back to you.

I'm a Certified installer for Trex decking. Trex Transcend is encapsulated board, with a finish that resists staining, scratches, mold and has a high definition wood grain surface. I don't know what you did to it or whoever worked on your house did something to it, but Transcend decking does not need staining or painting and the protective shell protects decking against fading. Trex Transcend and a few other Trex decking products like Trex Enhance, and Trex Select, they cannot be painted or stained. The only thing can be painted is Trex Transcend White Railings and you should only use light color paint. If you use dark colors, it will absorb heat and it will damage the surface and you can kiss your railings good bye. Also high quality fast drying primer should be used and you have to use Oil or Water based exterior paint and most importantly surface prep is the the key. When it comes to painting specialty surfaces like Trex Railings, hire a Pro to do the work. With that said, if you installed Trex yourself, you have to go to the store and have a Trex representative come out and check your deck. They will determine if loss of color fading was due because of negligence or not. If its a material defect, they will cover the material cost. If you had a Certified contractor do the work, they will cover material and labor. Call your contractor and he will get a Trex Rep to come out within a few days. Good luck

It looks like it will work as long as the mirror is attached to the wall......

There are lots of nice cordless lamps. What if you take the picture to a store that just sells lighting, and see if they have something similar in their catalogs? It would be expensive but maybe worth it if you love having them be the first thing you see when you come in the door.