Be careful when you buy the replacement shaft. There is a difference in the thread diameter between chain drive and shaft drive. Yours is a 14 mm nut. A chain drive is 20 mm. The thread pitch may also be different. I don't know for sure.
The Kawasaki Z1 service manual gives a range of torque value on the nut from 60 to 83 ft-lbs for 14 mm coarse thread (2.0 mm) and 53 to 74 ft-lbs for fine thread. (1.5 mm)
I'll preface my comments by saying that I'm a mechanical engineer who studied fracture mechanics in university, but that was waay back in time when the earth was cooling, and I never practiced fracture mechanics after graduation. I looked at the photos posted of the failure surface and it appears that the failure was under high stress rather than a fatigue failure. In other words, it was a one time event rather than the result of many cycles. It also appears to me that it failed in torsion, not due to bending. Possibly someone was doing wheelies or burnouts at some point...who knows. But it was likely something like that.
I looked at the fiche for the transmission on Partzilla to see how everything fits together. I don't see where there is a lot of lateral force on the shaft, which is probably why you were able to continue riding the bike with the broken nut. Everything inside the cases is held together by a snap ring on the far end of the shaft, along with a couple of snap rings between the various gears. The bearing cap on the far end is held by a small "button" in the case, so it can't move, but the bearing inside can move. The big bearing nearest the failed end is held in place by a retaining ring. So unless there is some unusual lateral force I can't see the shaft moving laterally. I think the main reason for the nut is to hold the shaft damper on the end of the shaft, and I don't understand what holds it in place now that the nut is broken.
As others have said, with proper tools repairs are not difficult, even for someone who hasn't split cases before but who follows directions carefully. You will have to pull the engine out of the frame, remove all the case bolts and pry the cases open. The transmission shaft should lift right out and you can then pull things apart on a bench. There are, however a few things not mentioned in the book, like chasing all the bolt holes, etc that you will not know about, so be sure to stop and ask for advice if you are unsure. My two guiding recommendations are to take lots of photos of things as you dismantle them so that you can go back and review how to remantle them. And to buy a bunch of baggies to hold the parts and prevent losing something important, which always seems to happen. Which ever way you go...good luck. It's a really nice looking bike!