Cannot connect to WiFi Broadcom Corporation BCM4313 802.11b/g/n Wireless LAN Controller (rev 01)

Please get a temporary wired ethernet connection, open a terminal and do:

sudo -i
apt-get remove --purge bcmwl-kernel-source
apt-get install linux-firmware-nonfree
echo brcmsmac >> /etc/modules
exit

Detach the ethernet, reboot and tell us if it is working.

If you suffer poor signal strength, a well documented bug: https://bugs.launchpad.net/ubuntu/+source/linux/+bug/856314; I suggest you try the brcmsmac driver backported from kernel version 3.11. Obtain a temporary wired ethernet connection, open a terminal and do:

sudo apt-get install build-essential linux-headers-generic

Download this file to your desktop: https://www.kernel.org/pub/linux/kernel/projects/backports/stable/v3.11-rc3/backports-3.11-rc3-1.tar.bz2 Right-click it and select 'Extract Here.' Now back to the terminal:

cd Desktop/backports-3.11-rc3-1
make defconfig-wifi
make
sudo make install

You may get a comment about a private key. Unless it is mentioned as an error, it may safely be ignored. Reboot and let us know the result.

If signal strength is still low, try an earlier bcmwl-kernel-source as mentioned here: https://bugs.launchpad.net/ubuntu/+source/linux/+bug/1110139/comments/42

wget http://us.archive.ubuntu.com/ubuntu/pool/restricted/b/bcmwl/bcmwl-kernel-source_5.100.82.112+bdcom-0ubuntu3_amd64.deb

Or if yours is a 32-bit system:

wget http://us.archive.ubuntu.com/ubuntu/pool/restricted/b/bcmwl/bcmwl-kernel-source_5.100.82.112+bdcom-0ubuntu3_i386.deb

Confirm 32- or 64-bit:

arch

Then do:

sudo dpkg -i bcmwl*.deb

Reboot and give us your report.