I think a self adhesive strip kit with slim transformer will be best. I'll order one, make the frame and mount the mirror then see how bright they are.
You have several problems on this project that will need solutions to - and I'll start by suggesting you've got the wrong LEDS
At some point, it'll fail, so it needs to be easily removable. As others have already said, you really need to transformer in the loft, so you need to treat it as a light switch so that you've got a feed down, and then a switched live going back to the transformer, but ideally, you'd have a sensor that switched the LEDs on. I just picked up a set of under cabinet lights from COSTCO for about £27, and they had 3 led strips, the transformer, a sensor to switch on and off and dim, but possibly not enough cabling for what you're trying to do.
If you had a normal set of LED strip lights, then you could encase them in a cover, but they don't turn though 90 degrees, so you'd need to cut them, and then reconnect them behind the mirror backing.
You wave in front of the sensor. How will you turn the LEDs you've got through 90 degrees, or would you be best just fitting them either side of the mirror. It's going to be more of a pain mounting the transformer in the loft, but one there, it won't get disturbed.
The tube is pretty flexible and I have a 50mm gap all round so I'm sure they will go round the corners.
I think I'd best speak to a sparky about the wires. I can't get my head round the transformers and switch. The way I imagined it to come on with the light would probably end up being annoying unless it's very dull.
If you need to find a tradesperson to get your job done, please try our local search below,
or if you are doing it yourself you can find suppliers local to you.
Select the supplier or trade you require, enter your location to begin your search.
Please select a service and enter a location to continue...
Are you a trade or supplier? You can create your listing free at DIYnot Local