Chances are the legs of the desk will not fit into the sandwich on the first try. Use a file to remove materials from both pieces until the legs slide in and out easily.
Filing is a major pain. Not only is it labor intensive and time consuming, it can damage the veneer. If I was doing this again I would have spent more time or gotten another tool to do the cuts to make them more accurate. More accuracy on the cuts would have make the filing portion a piece of cake.