There is a way to replace the broken axle without removing the rear axle housing assembly from the car. This is often done when the task is just to get the car back on the road. Were it mine I would remove the entire rear axle assembly from the car and completely overhaul it.
If you plan to do that go to
www.ocmafc.com and on the left menu scroll to "tech articles" and click. It will bring up a new menu. At the top is my name, click on it and it will bring up a list of articles. Scroll to the one on rear axle differential overhaul. You are welcome to download it and print it out.
If you want to just replace the axle you can spread the spring and remove just the left axle housing and then pull the axle assembly out from the left side. The axle assembly includes both axles and the carrier and ring gear. Take the assembly to a work bench and separate the carrier halves and replace the broken axle. Measure the thickness of the gasket(s) on the left axle housing and replace them with the same thickness to preserve the carrier pre-load (if one is actually still there).
Either way you approach this, first obtain a proper spring spreader. If you pull the entire rear axle assembly, leave the spring under the car, do not pull the rear out with the spring still attached as it is not a safe thing to do.
Good luck with your project.
Tom Endy