Inspect the damaged area. Make a mental note about the type of damage -- whether an area is cracked or completely missing -- and the size of the repair needed -- for instance, a small hole or damage to the frame.
Draw sketches of the weave pattern in the damaged area so that you can refer to them as you replace the reed.
Stop any cracking from getting worse by applying boiled linseed oil or tung oil. Apply a thick coat and keep applying it to areas where it gets absorbed. Wipe the furniture clean when it won't absorb any more oil, and allow it to dry for 24 hours
Soak the rattan where you will be making repairs by wetting towels and letting them sit on the furniture for about an hour. Soak any new rattan you will be making the repairs with by letting it sit in water for about 45 minutes.
Remove the reeds that are broken with a utility knife. You want to remove the entire reed all the way across the furniture and replace it, rather than just removing the reed in the damaged areas. It's best to cut the reed at the frame on both sides.
Insert the new reed into the space and refer to your sketches to recreate the correct pattern of the weave. Attach the ends of the reed with a staple gun.
Leave at least 2 inches at either end of the reed so you can tuck it under when the repair is finished. Allow wet reed to dry for at least 24 hours before applying finish or attempting to use the furniture.