That's an interesting floor. Is it cold to the touch? I ask because it looks rather quite a lot like actual sliced stones with grout instead of like any sort of resilient flooring with stone-appearance surface; the breakage all seems to have happened AROUND the "stones". I've seen sliced stone available attached to fiberglass mesh to make mortaring & grouting easy; it's laid similarly to ceramic tile.
If that's the case - that it's actual slices of natural stone, it may be really difficult (or at least impractical) to polish the surface with any sort of abrasives. Paste wax might add more shine, but at the cost of making the surface more slippery underfoot. Another possible option would be to pour a layer of acrylic resin (possibly "bartop finish") or decoupage over it, just deep enough to fully encapsulate all the stones. If you did that, you could even patch the damaged areas first with sliced stone (stones CAN be sliced with a diamond-rim wet tile saw) and grout.