The read/write head is essentially a tape-head, so normal cassette-deck cleaning fluid - which is usually isopropyl alcohol (or pure ethanol, if you have a source) should be fine. Don't put loads onto the fibre disk though, it should not be saturated - because then you have to wait quite a while for it to evaporate from the insides of the drive. Just a few drops on a small area of the disk is enough. The cleaning disk can be used quite a few times, as you only need to wet a small stripe or two to ensure that the head gets wiped a few times. I only have one cleaning disk, I've had it for about fifteen years, and I've only used it three or four times.
Aside: I stick an 850W vacuum with its smallest "slot" nozzle at the opening as a routine method of cleaning really dirty (dusty/hairy) drives. No problems as long as nothing inside is loose. Quite often the cause of bad reading is that there is a cat or dog hair in there, but most often the cause is a bad floppy disk and no amount of drive-cleaning can fix that - the cure is to use a better floppy. I also routinely use said vacuum technique for cleaning motherboards and the inside of computer cases. Be careful and you should have no problems - but don't come crying to me if anything goes wrong, I have never caused any damage at all by vacuuming my computers.
Computer cleaner:
