Your going to need a stiff broom, and possibly a mixture of water and TSP, to get that overspray off the floor. Scrub the solution in a small area first, to see if it will lift the paint off. Then wipe up, or wet vacuum up the slurry. Not sure of your set up, but spreading solvents on a floor can lead to hazardous fumes, which can enter your home or fumes being ignited by a flame source.

Knew a guy that painted his Mustang in his garage. Everything in the garage ended up with a fine layer of red overspray. Shovels, lawnmower, car parts, tool box, posters, books... etc. The paint job looked fine, the garage interior, not so much.

Tri-Sodium Phosphate, most hardware stores or paint sellers should have some for sale. It is a powder (or can be premixed), that when mixed into water can be used for cleaning paint brushes. Especially useful in cleaning walls before painting, or used to clean a surface after a solvent has been spread on that surface.

Again, not sure if the TSP will work, but it is not too expensive and your floor will be clean, but possibly still have some purple.

Yes, lacquer thinner will need to be used. You could use a lower grade lacquer thinner, some companies call it 'Gun Wash'.