A zoomed in review of the original installation reveals a potential problem that I would address. As the OP correctly noted, the cable is original cloth covered Romex, without a ground. In my experience, the insulation surrounding the individual wires in cable that's this old is often very brittle at this stage in it's life, and will crack and fragment off if manipulated, or bent sharply. The potential safety problem is that at some time in the past, some idiot installed the run by bringing it through the hardwood floor, flattening it out, and installing carpet over it, basically making an illegal and dangerous installation from day one.
At this point, I would work with the assumption that the Romex is damaged and needs to be removed. I would turn the breaker off, pull the wire through the floor, cut it back until you are past the questionable area, and end it in a new junction box. This can be as easy as putting it into a $0.50 blue nail-on style switch box, capping the ends again with wire nuts, and installing a blank switch cover to close the box.
Oh, and do yourself a big favor, ASAP. Grab a flat bar, and pull those scraps of carpet tack strip up. Stepping on those little nails in bare feet is unpleasant. Kneeling on it is an can be an experience to remember, as in laying in a ball, tears of pain, amazed that something can hurt that bad.
