Open Source and Free Software

It sounds like you've hit all the ones I would have suggested. Apparently this laptop doesn't support 64 bit?
Rather than frustrate yourself I would suggest getting a USB wifi card. I've had good luck with those in situations like yours. They aren't too expensive and can be found easily on Amazon. I have forgotten which brand/model I've used or I would point you to a specific one.
I did see one person on the Linux Mint forum say that they had to unplug their ethernet cable/card for this Broadcom wifi chip to work.
Its been years since I have had any issues in Linux with hardware. I guess I got lucky and never came across this chipset in my various systems over the years.