So, businesses that post videos of themselves cleaning floors like this seem to use a commercial, truck-mounted hot water extractor. These are ... much more powerful than anything easily available to a consumer, because they use strong detergents, near-boiling water, and very strong suction to basically wash and dry the floor in one pass.

If I was in this situation, I'd shop around a few cleaning companies that have a truck, and then see if they'd give you a discount for coming out 3-4 times a year on a schedule.... or I would look into sealing the tile with something clear but self-leveling so that you can take care of it yourself.

Keyboard cleaning slime is the same as the slime kids make at parties. I think it’s made of white glue and borax.

So at least it probably need not be too expensive.

You might be able to mix up 5 gallons or so, dump it on the floor, and sort of roll the slime around with squeegees.