Step 2: The line's pathway
The drawing shows the pathway the line makes through the holes at the ends of the frame. The left portion of the drawing shows the line (gray) and the frame (brown) as viewed from the side of the glasses near where the bows attach to the frame. The right portion of the drawing is a side view and shows the bends the line must make. The lens would fit against the left side of the view in the right portion of the drawing. The stubby end of the line is supposed to press down into the channel so it does not interfere with the fit between the lens and the frame.
See my Instructable titled "What gage is that steel or wire?" to determine the thickness of the monofilament line your glasses use. Or, try to match up the end of a spool of line when you are in the sporting goods department of your local store. It appeared 0.029 thick 50 pound test line should have worked, but it would not thread through my holes without a lot of work forcing it. I chose 0.022 thick 30 pound test line. A spool of 400 yards was about six dollars US.
Use a knife point to pry the old broken line out of the frame holes. Attach the end of the new line as shown at the end of the frame near the nose pads. Place the lens into the frame and pull the line over the circumference of the lens. Cut the line so it is an inch or two longer than needed.
Remove the lens and thread the end of the line through the frame holes near where the bow attaches. Before trimming excess line, place the lens in the frame and pull the line over the lens. This will show you how much you need to cinch up the line so it will be tight when the lens is permanently mounted. Remove the lens and adjust the length of the line between the ends of the frame. Keep working at this until the lens will be held firmly by the line.
Trim the line to length and try to press the stubby end into the channel. I was not able to get it to go into the channel, not even with a needle nose plier. My lens seems to stay firmly in place, anyway.