Rerouting a new ethernet cable is near-impossible.
It might be worth just trying a different cable just by routing it through the rooms. Cheap cable testers are available but usually need access to both ends. Some PC BIOS's (and maybe routers) have built in ethernet cable testers. If defective, the existing cable may just need a new connector. If it's damaged in an inaccessible area, there is little choice but to replace it. You may be able to tape a new cable to the existing one and pull the new cable in while pulling the old one out.