I would definitely go to acid. If you are cleaning tile on a regular basis you need to keep some tile acid in the truck. 95% of the time the acid will not be needed but when you run into the job like you had today it can be a life saver.
You don't necessarily need to use acid on the whole floor. I keep a dedicated pump sprayer with acid in it as well as a spray bottle. Sometimes you only need acid in front of the sink or stove, sometimes the whole floor will need it.
Just be careful not to get any acid overspray on any stainless steel because it will instantly stain it. Also make sure it's not a stone floor, test by putting a drop of acid on a tile (not the grout) in the corner, if the acid fizzes up then do not use acid on that floor.