Just get one or more of those battery powered, stick on lights. You won't have to worry about wiring anything up to the car.
I have several of those inside my gun cabinets. Even have one in my mini-fridge (fridge didn't have a light from the factory)

 

I still have mine from my 1994 Suburban. I removed it to fit some extra underhood accessories. The cord would reach all the way back to the rear tires.

Steve

 

the project is good, but ...
expensive, tinkering for a long time, the result is narrowly focused - only the engine compartment light, a little bit dangerous - interference with the wiring and additional consumption

An alternative option is a portable lamp in the glove compartment. I carry Cosmos with me.

Powered by batteries, equipped with a magnetic mount. It is easily attached under the hood, you can pick it up and shine in hard-to-reach places. You can shine into the spare wheel well, you can use it when looking for small things in the cabin or under the seats, you can look under the bottom of the car. I once used this to find a key that I dropped on a dark road.

I know a guy who carries with him several closet magnetic lamps - this one is generally cheap. And it shines well, and it's not a pity to lose.

 

thanks you!! I decide to go with led white strip for under hood and inside the car, to the inside car, I will connect to the door wire when the door is open the light will be on.

 

Get the harbor freight braun 18650 cob with built-in mini USB recharge. $20. I think home depot also has a husky version but unsure if 18650.