Andi, first of all if you've been asked to give him a good price because he says he'll give you regular work.... time to think whether that's actually the case or if he's just trying to work one one you :
Heard it a thousand times before....
As to what products to use, the first thing I'd do is soak the floor with a sanitiser. I'd even go and treat it once and leave it 24 hours before even going back to complete the work. What's been happening for the last 20 years is urine has been soaking into the grout. I'd even go as far as saying that no amount of cleaning could catch up with the amount of urine that's gone into that grout.
Anyway, after sanitising I'd probably give it a good dose of strong floor cleaner or even stripper, scrub the hell out of it with a black pad and then rinse. From that picture I couldn't say what the finished result would look like.
You'd probably need to follow it up with another clean using a phosphoric acid based product to get rid of the efflorescence and scale that's built up over the years.