I would be extremely surprised if you can flash the BIOS on a dead board... You must be able to boot to the floppy disk, and I do not see how you can do that without a BIOS.
Tell me if I am wrong.
A nine-year old motherboard sells for about $20 on eBay. That should be the fastest course to a resolution.
There are tricks to the floppy disk as well.
#1 you must use an original IBM Floppy disk... It cannot be formatted by any version of an operating system higher than Windows 95. Not Windows 98, WMe, 2K, XP...
Then you must install original IBM-DOS boot code to the floppy.
Then you must copy the BIOS update to the floppy that has the boot enabled without damaging the boot code.
This is explained in detail at several sites online that you can find with a google search.
The BIOS boot will not work unless the floppy disk is setup correctly, and an old Floppy disk may be too stiff to work or to receive that data.
Find a good site online, then follow the instructions EXACTLY.
Still, the motherboard must be capable of booting a floppy drive, and without a BIOS, that is probably not possible.
Worth a try, perhaps, but...