Thanks to Tiffani, we now are in the club of people who can see inside this dark cupboard. I bought two of these a while back but only just installed this one yesterday. It's the Dioder LED battery-operated lamp from Ikea ($14.99). It's got a sensor, and when you open the cupboard it goes on for 15 seconds and when you close the cupboard it goes off after 5 seconds. It lights the cupboard all the way to the back, which is awesome.
I put one on the top of the closet too. Im too short to see to the back!