Adding an LED light is relatively easy; you need an LED, a battery, and a switch to turn the LED on and off (sometimes a resistor to reduce the battery's current so the LED does not burnout too fast,). All of these parts are provided in those small keychain lights. I would place the battery and LED (and resistor if necessary) into the endcap which you want to light. I would place a simple contact switch in the center grip and run small wires from the center grip to the endcap. In the center grip there appears to be two, semi-circular areas almost made to place your thumb. That could be ideal to place the contact switch. If you want to have the switch also in the endcap then the whole situation is much easier, you could use the entire light almost "as is".
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Here are photos of a partially and fully disassembled very inexpensive light which is often given away as a promo/advertisement for free. However you can never find one when you need one for free so you can also pick them up on ebay. I bought several as the shipping from China can be more expensive than the light. (Often the light is as inexpensive as the batteries alone.) You see just how simple you can make the circuit. (The fully assembled version is at the top of each photo with the keyring still attached.) Depending on the sizes you might be able to reuse the entire packaged light and just "move" the switch by taking it out of the package and replacing it with something similar, either also in the endcap or in the center grip.
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This design places the batteries side-by-side (with just a thin piece of metal connecting them) to provide a suitable length to fit between your fingers. Placing the batteries on top of one another will make the package thicker but shorter which might fit better into the endcap. It is difficult to do much damage to these and even if you destroy them you are not out much money so experimentation is painless.
Any thin wiring needed can be taken from almost any old telephone/computer cable.