It's possible that the drive is not configured properly, or that you have it plugged into the wrong connector on the cable, much like IDE hard drives with 80 conductor cables used in newer systems old floppy cables have a 'master' or 'A:' connector and a 'slave' or 'B:' connector, typically the connector closest to the end of the cable is the 'A:' connector and the one near the middle of the cable is the 'B:' connector, however the drive very likely has a jumper block or set of DIP switches that may have to be configured in a certain way to specify which set of signals the drive should accept depending on it's positioning on the cable. Bear in mind also that these signals are 'flipped' for each position on the cable by means of a 'twist' in a few of the individual wires in the cable in order to send the signals to the 'A:' drive.
One thing you can try is to try connecting your drive to alternate connectors on the cable, it's possible that your drive is already jumpered to think it's a 'B:' drive in which case if you attach it to the 'B:' drive position on the cable it will act as the 'A:' drive in the system possibly fixing your problem in a manner.
Additionally are you absolutely certain that your drive is a 360K drive and not a 1.2M drive? The two will look almost identical (unless your drive is a full height 5-1/4 drive of course), but if the BIOS doesn't support the 1.2M drive you might get errors like the ones you are receiving possibly. Additionally you need to watch out for the fact that one of the 2 cables used with MFM/RLL hard drives is nearly identical to a standard floppy cable aside from the fact that the twist in the cable involves a different number of wires.
And of course your I/O error might be a result of the settings of the jumpers/DIP switches on the drive, especially if you tried changing any of them to get things working and possibly didn't get them set back to the correct positions.
Hope this helps,
bartman2589
PS - I thought I should clarify one thing, it is possible to make a floppy drive cable that does not have the little twist in it as long as it is being used with drives that have the capability of being jumpered to specify whether they are a 'B:' drive or an 'A:' drive, not all drives have this capability and in the interests of simplifying manufacturing many companies that make drives stopped providing this capability as making a modified cable that would simply swap the signals was more cost effective than integrating the circuitry to actually accomplish the same task when a jumper or switch was changed. This is most notable on more recent 3-1/2" drives, many of them do not have any jumpers at all and instead are configured to a 'default' specification to work with cables that have the twist in them.