I recently saw someone do this on YouTube, so I bought a Jenga from a local charity shop and got started.
Using super glue and wood glue I stuck a few of the pieces together for the bottom where the IKEA pre made lamp holder and cable will sit. Bit by bit I put the lamp together cutting a few pieces and drilling to make room for the lamp. I had to have a solid base with a couple of dowels fixed in upright and the 2nd part of the tower with corresponding holes for the dowels, this allows lamps to be replaced.