Floppy support was removed for Socket 1156 as it was first with a
PCH instead of the previous
ICH10. As many industrial machines run off of floppies or use ISA cards, newer boards with extra super I/O controllers or bridges to ISA slots continued to be made, but industrial motherboards tend to be rather expensive. However if you find a board with an ISA slot you could try an old ISA super I/O card intended for a 386 or 486. Every one of those needed such a card because there was no onboard support for floppy or IDE drives back then--just be sure not to get a VLB one.
It's too bad ATAPI support was removed back with ICH9 because one floppy drive--the SuperDisk LS-120, plugged right into IDE. Or an IDE Zip drive would also fit in that bezel. You could try one of those with a IDE-to-SATA adapter. Or they also came in SCSI variants so you could get a PCI SCSI card.
Clearly, the problem is the case isn't old enough. One with 5 1/4" floppies would let you put slotload Blu-ray burners behind the cool latching levers.