Hi Phil, had a look at the pictures you posted simple enough problem to solve.
I would pre spray the tiles and grout lines with Bio-Kleen heavy duty degreaser diluted at 5-1 for deep cleaning, then let the solution dwell for 2 to 3 minutes, first scrub in between the grout lines with a grout cleaning brush to loosen all the ingrained and impacted soils, then when all the grout lines have been scrubbed proceed to scrub the tiles with a microcleaning mop pad, these wonder pads will deep clean the tile surface while at the same time absorb all the loosened soils you dislodged with the grout cleaning brush.
Finally use damp waffle microfiber floor cloths to flat mop the floor to achieve a streak free finish, Judging by the photos Phil i woud think your customer was using traditional cotton mops on a daily basis to clean the floor and that is why there was such a build up of soils in the grout lines. You see Phil the fibers in cotton mops are round and thats all they are capable of is pushing the soils around the floor, so when the moisture evaporates a light film of soil will dry back on the surface of the tile and into the ground lines, this problem is compounded on a daily basis as each time the floor is mopped with a cotton mop there is another light film of soil going to dry back on the tile and into the grout lines, the proof Phil is in your picture as the grout lines have filled up and over from daily mopping with the cotton mops (one of the most useless tools used within the cleaning industry) and you can see quite rightly Phil why they fit perfectly into the CFU category of tools used within our industry.
Recommend your customer to make the change from traditional cotton mops to waffle microfiber floor cloths they will save them alot of money as they are extremely durable and can be laundered up to 500 times.
Kind regards
Tadgh