Bleach in itself isn't a great cleaning agent for these purposes because bleach doesn't remove grime; you can try bleaching grime but the end result will be partially-bleached grime.
For flooring (lino? But probably vinyl tbh...) either the surface has been sorta etched by dirt and discoloured by staining which has penetrated into the etches, it has a wax coating which has incorporated dirt into it/which needs to be stripped, or it really is just the pattern.
I would get hot water, dish soap, and a proper traditional mop rather than any sort of sponge or flat mop. If it's really bad you could get a floor scrubbing broom also.
With a sparing amount of dish soap and plenty of hot water, clean the floor but change your water when it gets dirty. Don't wait until you have finished half of the floor or all of the floor to dump out the water - as soon as it starts looking grubby, dump it and change it.
On floors that do not get cleaned properly, you often find yourself changing the water every couple of minutes and tbh if it's a big job or it's really grimy, you will likely want to run two buckets - one filling up at a slow-ish rate and another that's being used, rotating them out as you go. This can also be done using one mop bucket and one regular bucket, just use the regular bucket to fill up the mop bucket when changing the water over. If it's as dirty as you say, be prepared to mop over the same patch or room multiple times until the water starts coming up clean.
If it's waxed and they've done a bad job by waxing over a dirty floor or using an all-in-one floor cleaner with wax and it's accumulated over the months/years then you're probably going to want to consider going in-depth and possibly getting a specifically designed floor wax stripping product. You should consider re-waxing the floor, especially depending on what flooring you have.
Moving forward, flat mops or sponge mops are for spot-cleaning only. Make sure that you use a real mop when you clean your floors, and always follow the above advice about changing the water over.