They make LED rope lights which are embedded in plastic and are often 230V. With these, the LEDs are difficult to aim since the rope twists. Hey are also not particularly bright.
They also make LED strip lights which have adhesive backs and operate on 12 volts DC. You probably want this. 12 volts is safer for the method you want to build this. Brightness is not a problem with these - with double density (120 lights per metre, 9.6 watts per metre, triple that for RGB) you will have more light than you may want. These strips can be cut every 25-100mm depending on density and voltage. Examples: regular 60/m ...waterproof 120 LED/m... RGB sleeved, addressible
I would not even try to buy these locally. Any local seller will have very small selection, and/or very high prices. You want the large selection you can get on the Internet.
These systems are custom by nature. Not terribly hard to hook up, but you need to learn the craft involved. You would need
- a 12 volt supply of appropriate size
- if color strips, an RGB controller; if white, a dimmer if you want one.
- amplifier(s) if using a controller or dimmer
- appropriate wiring methods to attach to the strips at as many points as possible.
To install them: they typically have adhesive backs. You cut sections to length (at the designated cut lines). They must install straight: they can bend, but only in one direction (in which they are spooled). They make variants where the LEDs emit out the edge instead of out the top, if that helps.
Make sure the surface is sealed and dust-free. For instance if you tried to adhere it straight onto plywood, the adhesive will probably let go.
If you're going on bare wood or other surface, rather than painting, try just laying a thin sheet of metal e.g. aluminum, they sell it at hardware stores. I wouldn't use steel, it will rust. Screw that into the wood with short screws, clean/dry it, and the LED strips will adhere nicely. This will also help pull heat away from the strips, though they don't get particularly warm unless they're lit still-in-the-spool.
To wire to them, there are little solder pads on both sides of the cut lines. You can either buy slip-on connectors which will tap those cut lines... or my preference is to solder 20-22AWG wire to the pads. That is enough to handle the current of a metre or three**, but thin enough to be flexible - if you use wire that is too heavy, "the tail wags the dog" - the stiffness of connection wire pulls on the delicate strip and could potentially warp or damage the strip, or lift it off the adhesive.
** Rule of thumb, when AWG goes up by 4 units, ampacity is cut in half. 12AWG=20A 16AWG=10A 20AWG=5A etc.