I permitted myself to re-add an idea from faze one (I don’t know if this is ok) maybe you could use some of its features. I believe it could be a solution for the power supply and in the same time it adds another dimension …volume … to the shape of the lamp offering interactivity too.

This lamp is composed from several LED light bulbs special designed so that they can be joined together like in a LEGO game … and an universal adaptor plug for use instead of a normal light bulb. This is like a construction game … you can make from this modules different shapes (objects/compositions).
As long as they are directly connected to a socket or by other modules have contact with electric energy they are charging. After the LED bulbs are charged you can separate them and use them as candle lights (spread them thru the room to create a special atmosphere or just for reading) … or crate separate composition of lighting objects.
This LED bulb can take various colors … every unit (construction module) act as a control (switch) for the rest in the shape you created out of them so that you can set the light color for all elements in a composition or make them change color in time. The LED light bulb could be any kind of reg. polygonal shapes.