The necklace in question
The offending clasp
This clasp is called a screw clasp and works exactly like you would expect, by screwing the ends together. Of course, in a necklace, you can't see the clasp so getting the ends together is a challenge. To make matters worse, the plastic necklace string was tied off in the clasp and tended to lie on the threads so I could never get it to work.
A better view
A sensible toggle clasp and new necklace wire
Cutting the original string gave me some beads to start with. I tied the new wire on the toggle in a square knot and threaded a dozen beads onto the wire. The terminating end of the knot I also tucked into the beads so it wouldn't bother me. This secured the wire end in the necklace.
Restringing the necklace went well and I only lost two or three beads.
I tied the wire off on the other toggle end and strung the wire end back into the beads. The beads aren't completely flush with the clasp but that doesn't concern me.
With a toggle clasp, I have finally wore this necklace.