I don't know if this will help or not: I got a small drill-snake stuck in the drain for my laundry (under the slab) many years ago. I chucked the broken piece of cable into an old Bit-And-Brace type drill and turned it counter clockwise. It snapped free.
The second time I broke one in the same drain (5 years or so later), there was no auger cable above ground. This time, I was able to rent a small snake with a retriever on it. That worked well. (Send it in clockwise until it just stops. Turn it by hand a little more & pull like a crazy person.) The retriever is a stiff wire bent into a shape of a spring that starts wide (end) and narrows (cable end). It wraps itself around the broken cutter.
In my trade, I use a retriever sometimes in large underground cable vault conduits to clear out melted rubber insulated 1920's era wiring so I can re-use the conduit (usually concrete or clay tile- Sometimes even beautiful, thick walled brass.)
Next time that drain line clogs- Out comes the Bosch, like Chris said! Good bye Galvanized, Hello PVC.
Hope one of these tricks saves you from having to clean up from breaking the floor.
(Clean up is the worst part, right?)
Paul