We rehung the shade with substantially less trouble, having figured out the secret incantation at 59 minutes into the hour we spent doing it the last time. Gretchen pulled on the string and it glided majestically to the bottom of the window, straight as an arrow.
If only the upper part of the string had stayed tucked into the little plastic bead that joins it to the bottom part of the string.
I called the store. A repair person will call on Monday and make an appointment to come out and put the string back in one piece. Gretchen tried tying it off, but the inch you lose that way is enough that the shade won't reach the bottom of the window.