Generally speaking, unless the device you are putting it into requires it, do not format anything in FAT. The FAT file format is prone to getting discombobulated, i.e. really messed up. Instead format FAT32 if you must, NTFS, or best still exFAT which can be read by Mac, Linux, and Windows. Of course test any new formatted card in your camera BEFORE expecting it to actually work.
Your SD card is not likely really broken. Even if Windows cannot currently format it.
A handy, free app, by the SD Association, can be found at
https://www.sdcard.org/downloads/formatter_4/. The app there can rescue many
broken SD cards. I believe it re-writes the firmware of the SD card as part of its formatting. Regardless, it formats many fudged up SD cards that Windows, et al, can't.
Needless to say; but I will anyway, DON'T use
SD Card Formatter unless you are willing to part with the data on the SD card.
Generally speaking, unless the device you are putt... (