To find out which PCI.ID you have, we proceed to opening the terminal by pressing CTRL+ ALT+ T (It should open a window with a blank background) and inside this terminal we run the following command: lspci -nn -d 14e4:
The key to finding the correct driver for any network card is what is known as the PCI ID (PCI.ID). There are dozens of Broadcom wireless cards and more seem to appear every day. Knowing what Broadcom Wireless Card you have So with that in mind, the following is what we have right now which is simplified in just 3 steps: 1. Link 3 - Gives an error similar to "Sorry, installation of this driver failed."
After you follow this guide, you will NEED to test your wireless connection for at least 2 hours (I actually recommend 8 hours) with another device in either Ad-Hoc Mode, Infrastructure Mode or Both. In total we wanted to offer an answer that could be easy to follow and covered most Broadcom Cards / Drivers. Special thanks to chili555 who helped in the Ubuntu forums and on this site with many questions related to Wireless devices and to others who have contributed through E-Mail, chats, IRC and more in testing various drivers with several of the most popular Broadcom Wireless cards (Huge Thanks to Chili555 really. This answer is based on an extensive research done by various Ubuntu users that worked together in almost all issues related to Broadcom.