Trace the source of the damage. It is likely a leaky roof seam. If the roof does not need to be replaced, seal the seam. If a section of the roof does need to be replaced, sealing the roof will become the last step.
Peal away and remove any wallpaper or molding on the ceiling.
Cut out the damaged area of the roof with the jigsaw. Drill a hole to get the jigsaw blade into the ceiling. Depending on the extent of the damage an entire section of the ceiling may need replacing. Look at the molding locations to locate screws or nails underneath. These parts may also need to be carefully pulled away from the walls if attached.
Remove all damaged wood, starting with the main section of plywood and then the under beams. Using wood screws replace whole sections or cut into the good wood and cut the damaged wood out, and splice in new wood. This can be done by overlapping a piece of wood over the cut joint and screwing the new wood and old wood into the overlapping wood. Metal attachments can also be used for splicing. The stronger the bond the better. Gluing the wood joints will help the bond.
Reattach the wallpaper, molding and fixtures.
Seal the roof with sealant if it hasn't been done yet.
