Today I had a few minutes to kill so I installed LED lights in the closets
I bought self adhesive LED light strips from some nondescript company on Ebay. But I see these "light strips" everywhere so they shouldn't be hard to buy.
The light simply runs from top to bottom of the closet, through small holes drilled in the shelves.
A small "normally open" push button switch turns the light on when the door is opened
The light they throw is perfect for these closets.