Join these two objects together? I'm trying to join the bottom of this arcade stick to the face below it by actually having it attached to the mesh. What would be the best way of doing this?

I've tried to bridge the bottom edgeloop to the face however this did not work. I've been looking all over google to figure out how to do this but my blender terminology is next to non-existant so I wouldn't know exactly what key terms to be searching for to begin with.
Try to use the boolean modifier set to add and then fix the topology manually using the knife tool