Okay, just got mine working. Not sure if this will work for anyone else, but I played with the sort-of-working USB cable, straightening and unstraightening it to change its electromagnetic geometry, and then I plugged and unplugged the connector at the Leonardo several times. Finally, after the 3rd unplug/replug, my Win 7 laptop recognized it as a Leonardo and allowed me to upload a simple modified blink test sketch. Works fine now!
Just for fun, I tried my original "bad" USB cable, and it still doesn't work. Power is delivered to the Leonardo and it lights up just fine, but the PC doesn't recognize that it's been plugged in at all.
Seems like the Leonardo is very sensitive to having just the right USB cable.