The driver you currently have in use is known to have issues with this particular card that you have (BCM4313). For some, the older version of this driver (wl) works, for others the native brcmsmac works better.
Regardless of the driver in use, there are a few common suggestions to improve your connectivity -
1) Turn the power management off on the wireless interface -
sudo iwconfig eth1 power off
2) Change the encryption type in your router from WEP to WPA2-PSK (AES)
Try these first and see if you can connect. Although given this combination's reputation I doubt if it is going to help, but is worth a try. If it still can't connect, please try removing the current proprietary driver and use the native one instead -
If this doesn't help either, run the wireless_script again and post back the new report.
sudo apt-get purge bcmwl-kernel-source broadcom-sta-common broadcom-sta-source
sudo modprobe -v brcmsmac
There is another tested workaround that has worked for many, but I want to try the native one first : http://ubuntuforums.org/showthread.p...9#post12629619