You may be able to pull the sending unit out and work the float lever back and forth if it's stuck. You should be able to plug it into the wiring harness while you change float position and see if it is registering properly. Some carb cleaner may help. It's been my experience though, that once they crap out on you, they are hard to revive. You may be able to find a replacement, ebay has one listed for about $73.00 right now for your '01 zx6, part number 52005-1105. Gasket is about $15.00 on ebay, p/n 11060-1337. If you do need to replace the float, you may want to check the condition of the gasket and replace it at the same time if it's starting to crack. Also, once a bike sits that long, you may be facing some tank corrosion/rust issues, too. Not sure about hearing the fuel pump running, though.
Last edited by WeBeNinjammin'; 04-14-2012 at 08:56 PM.