Broadcom network card doesn't connect to wireless network

[D. Schetselaar]

Thanks for your replies. I (think I) did exactly what you said, but it did not solve the problem.

PCI.ID  = 14e4:4312

Then I did:
rfkill list all
the card was not blocked.

Then i did:
sudo apt-get remove --purge linux-firmware-nonfree
sudo apt-get remove --purge firmware-b43-installer

to remove the packages that I previously installed.

Then I did:
sudo apt-get install linux-firmware-nonfree
Reboot
Didn't work

So next I did:
sudo apt-get remove --purge linux-firmware-nonfree  (is this what you mean by starting with a clean system?)
sudo apt-get install firmware-b43-installer

reboot

Now it said: wifi disabled by hardware switch (or something like that) so i pressed fn f2 and the wifi light came on, it finds the network, but still doesn't connect

rfkill list all shows that the card is not blocked.

Anything else I can do?