How can we repair cracked and broken concrete around a sewer stack in the basement floor?
The right way to do it is dig out and repour the concrete in the area.  This isn't that hard or expensive.  

If you want just a stopgap until you decide what you are doing with your main line then you could get a bucket of hydraulic cement and patch over the existing concrete.  The cracked concrete should not be affecting your stack - hopefully - so this is really a different issue than your plumbing problem.