Soak the repair strands in a pan of warm water for 10 minutes. This makes them pliable for weaving. Keep the pan of water and a rag or sponge at hand to moisten the strand if it begins to stiffen while you're repairing the chair back.
Brush the woven chair back gently to remove any debris. Clip off any broken wicker, reed or other material on the woven chair back with sharp scissors or garden shears. Remove the remaining parts of the broken strands by gently pulling them free from the chair back, removing them completely. Tuck the free ends left from the broken strands into the weave so they don't show. Use a toothpick or awl to remove strand fragments clogging holes in the frame, if the chair back has holes bored in it for attaching the weaving.
Weave a matching reed or other material into the chair back following the same pattern as the undamaged areas of the weaving. For example, a horizontal strand might go through a hole in the frame on one side, across three vertical strands and then under the fourth vertical strand and repeat that pattern across the chair back.
Complete the pattern all the way across the chair back to replace the broken strand. Clip the end of the new strand 4 to 5 inches from the chair back.
Wipe it with a dry rag to remove any excess moisture. Apply extra-strength glue to the strand and tuck it into the weave, taking care to press it against an underlying strand to hold the new strand in place. Repeat for any other broken strands on the woven chair back.