I've had good success burning the tops of plants with clear plastic if the area gets enough direct sun. It does take a few days, but it works. I wouldn't use glass either, even tempered glass breaks into sharp pieces.
Once you kill the tops, you still are left with the dead plants to clean up. With the Jerusalem Artichokes, you should definitely dig up the tubers or they will most likely spring back up.
If you kill the plants without chemicals the tubers would still be edible. By all means offer to help your neighbor with this if she will share the tubers! If you like them you might want to grow some of your own (in a contained area, of course).
