In small tile areas like this we usually begin by hand treating and scrubbing around the toilet, behind the doors, and along the baseboards before moving in with larger equipment.
Because this tile was heavily soiled starting out we used a high pH cleaner but made sure to neutralize it afterwards so no residue is left behind.
We also finished by sealing the tile and grout to make future cleanup easier by not allowing dirt and spills to penetrate the tile and grout.