Good cleaners for a dusty, rusty cellar?

So, our cellar’s a dusty, rusty (but luckily not musty) mess. This wouldn’t be such a huge issue, but it’s where our laundry’s located, so the dust gets tracked around a bit. There’s a cement floor, but the bricks were crumbling, so we had many of them replaced and repointed, but the contractor missed a lot of rubble and dust when they “cleaned” up. Compounding factors, we later had two steam risers leak, creating a substantial amount of pink “rust” on the floor, in roughly a 5-foot circumference around the pipes. We’ve tried shop-vaccing and mopping, but the mess is pretty stubborn… so I was hoping somewhere here might have a recommendation for a more professional solution.

Eventually, once everything’s cleaned, I plan to paint the floors and limewash the walls– but right now, I just want to clean up the dust.