To confirm, you do have a eth0? If so. I highly suggest making sure your linux distro is pointed at the new(ish) Repo's https://docs.kali.org/general-use/kali-linux-sources-list-repositories and then make sure you can hit the internet through eth0 and run a full apt-get update. Reboot and try again. If that doesn't bring up the wireless NIC, there is a bigger issue. That broadcom NIC is pretty standard, so it should show up.
It is possible for OS images to get corrupted. Not common, but is a thing. I'd trouble shoot first with a lightweight distro like Lubuntu. Live boot that and see if wlan0 comes up. If so, try a fresh Kali image and copy the settings from lubuntu as they are pretty similar OS. If none of that works there is most likely a hardware issue. Also, don't forget a couple things:
Putting your wifi nic into Mon mode using airmon-ng and other things shuts off the network manager (most of time, depends on how you do it). So if it ever worked and you're doing some wifi cracking, that could be your problem. Also don't forget about:
iwconfig -a
Good luck, let us know if any of the above works.
