First you need to make sure the cable is fitted correctly.
It only goes one way round. One ond on the motherboard, the other end, nearest the split/twist in the cable, to the floppy. If there is a middle connector on the cable, don't use it, as it will not work.
Then ensure pin1 on the cable goes to pin1 on each connector - you may have to actually look at the floppy to check that end though the motherboard usually only allows you to put the connector in one way round. Pin1 on the cable is at the end where the red stripe is along the edge of the cable.
Then you need to check the BIOS to ensure the correct type of floppy is selected for driveA - this should be 3.5" 1.44MB. If it's wrong it won't work.
Then check that if the BIOS has a "swap floppy drives" or similar setting, it must be disabled.
Double check the power connector to the floppy is fitted correctly - it's very easy to push them on wrongly.
Then if all else fails, try a different floppy drive (Isee you already tried different cables).
If it doesn't work after this - and you've done everything correctly, then the only thing left that may be faulty is the motherboard.
If you want, you can also try resetting the whole BIOS to "Fail-safe defaults".
Note - the floppy drive does not auto detect in the BIOS - you actually set the BIOS to the correct type of floppy, then it should work.
You're not by any strange chance using an old 120MB floptical that runs on a hard disk cable are you?