01 Front end Swap

Your best bet would be a complete front end (wheel, forks, triples, calipers) from the ZX6R. You will probably need to have your steering stem removed from the 9's triples and installed on the 6's triples. Probably the easiest way to do it all.

I would think this may help answer it.

The '01 ZX9R part number for the part in the picture is 44037-1417.
The '03 ZX6RR part number for the same thing is 44037-1442.

I would say no it will not work, but I don't have the actual dimentions, just the part numbers. The parts can be identical for the section that runs through the head/bearing but be different on the lower triple that holds the forks.
IF it works, you are best off with a compete front end including rim, rotors, brakes etc. That would be the simplest swap.

I can't find part numbers on the bearings so check with Kawi, a parts guy can likely tell you if the parts are the same and maybe even the sizes of the bearings.

Don't look good though.:shake

It might be made to fit, but I doubt the lengths are the same or the angle in the tripples so it will probably handle a fair bit differently if you make it work.