It's quite the conundrum isn't it? You can't connect to the internet yet you can't enable the Broadcom driver unless you are connected to the Internet.
Set up you smart phone as a hotspot (a.k.a. as tethering). If possible, specify Cable Connection when setting up tethering.
You must have installed Ubuntu already on your laptop. With laptop switched off, join your smartphone and laptop with appropriate USB cable. Switch on your laptop and launch Ubuntu, you know, the one that won't connect to the internet.
Test whether the laptop is accessing the Internet via your phone by launching an Ubuntu app or by opening the terminal and pinging a server, e.g.
$ ping www.adobe.com
If the server returns test packets, type Ctrl-c to stop ping.
1/ In Ubuntu, launch Software and Updates, and click the Alternate Drivers tab. Wait until the proprietary drivers are listed.
2/ Then click the radio button for the Broadcom driver.
3/ Click the Apply Changes button. Wait until the progress bar finishes.
4/ Exit Software and Updates.
5/ Disconnect your smartphone.
Now select your usual wifi router/modem and enter its password. Then try pinging a server again. If it returns test packets you are done!
