The Foredom people responded promptly to my question about parts replacements. Meanwhile I was curious what my local small appliance repairman, Don, would have to say about the broken wire covering. Yes! We still have someone in our city who will repair things! In about thirty minutes he had shortened the wire by about two and a half inches and had re-attached the same connectors. When I put the hand piece back together and plugged it back in, it worked beautifully! I'm back in business, and at far less cost than buying new parts or a new hand piece. All that was thrown away was two and a half inches of braided copper wire.
Having spent a day back on the block with my hand tools, I have a different perspective on working with the power tool. I realize how I miss the control and slower pace of the hand tools. I believe I will find myself distributing the work between the power tool and the hand tools a little differently from now on.