I believe that is the Broadcom BCM4322. I would think Mint would have drivers for this.
So, how do you get internet without Wifi?
Do you have an ethernet connection you can use temporarily?
description: Wireless interface
product: BCM43224 802.11a/b/g/n
It should only ask if elevated permissions are required ... such as with apt. And it should. You can go into an extended elevated permissions session, at your own risk with sudo -i.P.S How do I prevent Linux from continually asking for my password.
sudo apt install broadcom-sta-dkms