The Broadcom 14e4:4315 is quite unusual in that the solution changes from earlier Ubuntu versions to later. In Ubuntu 13.10, which you are using, the device uses the native b43 driver and firmware that needs to be downloaded separately. Please first try to undo any possibly failed previous attempts; open a terminal and do:
sudo apt-get purge bcmwl-kernel-source
If the terminal reports that it was not installed so not removed, that's fine, please continue. Get a temporary wired ethernet connection and do:
sudo apt-get install linux-firmware-nonfree
Detach the ethernet, reboot and let us hear your report.