When you say you installed a new driver, was it a new version of the previous driver? Or was it a completely different driver/module?
The reason I ask is because there are a number of Broadcom drivers available, so it could be that you installed a driver for a different Broadcom chipset. So the driver you installed might not support the chipset on your wireless card.
Just a thought!
What output does the command "lspci -vnn" give you for the wireless card?
Note: The command will actually give a ton of information about all pci devices, not just the wifi card, so you'll have to scan through the output and find the section relating to your wifi!
Once you've located the output about your wireless card, it will have detailed information about it, including the driver/module that is currently in use.
Make a note of the driver which is in use, you'll need to refer to this in a minute!
As Kali is Debian based, it might be worth taking a look at the Broadcom wifi driver page on the Debian website and take a look to see if the module currently used by your card is listed there and check which models it supports.
https://wiki.debian.org/bcm43xx
From looking at the above, if your card is a BCM4313; you should be using the brcmsmac driver. Instructions on how to get that running are here:
https://wiki.debian.org/brcm80211
Don't worry about the step involving adding the Debian non-free repository, just make sure that the non-free repositories are enabled in Kali - Offhand, I think it should be enabled anyway! So it should just be a case of using apt-get to install the firmware module (firmware-brcm80211) and use modprobe to start using it!
Hopefully, ensuring that you have the firmware-brcm80211 installed and in use will fix the problem. If it is already installed/running, then I am afraid I am out of ideas!
