The old pull-switch was installed when we bought the house, and it was of a different type than the new one I bought. I went to replace it, and found out I couldn't just swap the wires, since the old one had an extra connection. The light switch box has this extra thing attached, and hooked in somehow, and I don't really know what it's for, or how to attach my new pull switch. My new pull switch comes with instructions to attach the black (hot) wire to its brass screw, and the white (neutral) wire to the silver screw. In the first picture, you can see the black wires are currently have a wire nut on, while the white wires are twisted around themselves, and exposed.
Attaching the new pull switch would be straight forward except I don't have a clue what to do with the extra black wire coming from that other bronze box.
Here is an overview of the light box in the utility room in the basement: You can see the bronze coloured, "extra" thing to the left of the box. In the bottom right corner, in addition to the regular electrical wire leading away, there are two smaller wires that originate from this box. I don't know where they go, since the wires are hidden once they go through the joist.
Here is a close-up showing the extra wire coming from the thing attached to the box:
Finally, here is a different view of that bronze box attached:
I'd appreciate any help someone can give (and any light you can shed on that attached bronze box).