Assuming your serious here. I’m not sure where you got 3 volts from. That’s rather high. Unless you’re runnng a very long strip. Most led bars (strips of pcb wire board with leds on them) are usually 1.25 or 1.5 volt feed.
Informing you up front I never did this exact build. You’ll need fairly low end led back light strips from a low end television. They have three wire etches along the board row. Occasionally 4. If you see five or more, or a single long feed etch that breaks off into each led, this won’t work. Those are addressable location boards and use a controller. In the dormant off state they are all off. Not all on. Meaning simply plugging it in to the board in will accomplish... darkness.
In the most direct method you’d need some medium gage wire, a battery terminal bracket, solder, and a AA or AAA battery. Strip the ends of the wire. Solder, staple, or otherwise mount the battery terminal safely to the back of the board. Solder the wires from the terminals to the corresponding lands on the board. Pop in the battery and enjoy single intensity lighting.
If you want to get fancier you can add (or make) a variable power dial between the battery and the board. Allowing you to adjust the intensity of the light by turning the dial up or down.
You can also use mains power if you add an AC/DC converter, but now we’re adding weight and bulk and making things more difficult.
Honestly though, you’re reinventing the wheel here. Go to the hardware store and buy a portable led pocket flashlight. They’re a cylinder a few inches long. About the thickness of a quality machine made cigar. They have 4-12 leds on one end and a button on the other. Almost always made of aluminium. They take 1-2 AA, 2-3 AAA, or 2-4 AAAA batteries. They cost 99č-$9.99 depending on extra features. Such as... some have multiple colours. Depending on the number of time you push the button. Or can change the intensity: 2 to 4 setting usually.
This is exactly what I described you try to build. A PCB (more accurately a PWEB) with leds on it. Attached to a battery terminal. With a switched fuse running through the button with wires attached. You’ll spend more doing your own than buying one in materials.
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