How bright would you want the lamp to be?  If you want it bright, two of the light strips would work well - they're flexible so you could bend them so that you have one led pointed towards each of the four sides.

If you want a dimmer light (night light?) a single pico light could work.

Hi Julie!  Thanks for your post, and thanks John Gerlach for your reply.  That's exactly what we were going to recommend.  Either a 2-pack of light strips or a 2-pack of Pico LEDs would work.  To power everything, you could add a USB cable and use a spare iPhone/iPad type charger to power everything from wall power.

This looks like a really cool lamp design!  Please keep us posted on how you light it up.