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Pop in a proper tiling shop (not Homebase etc) and have a chat. There are loads of really good cleaning products, some more corrosive/aggressive than others. You want something that won't damage the stone, whilst lifting the greasy gunk and left over soap etc, that will have built up. Once it's clean (and I mean CLEAN!) I can recommend a product called 'antidrop', which depending on how you want it to look, can either be Matt, satin or gloss. I like satin finish on those...I've fitted hundreds of metres of natural stone, as most tilers steer clear of it, for the reasons you've discovered!
I'm not having a pop at your cleanliness by the way, it's just that it needs to be very clean before sealing, as the finish is better!
The tile looks like pretty common indian slate. The issue was probably a void underneath, not properly back buttered before laying. The stone always comes in different thicknesses etc, so it's important to have 100% coverage to stop that happening.
Best option would be to grab a couple of spares from a tile shop and pop those in after a thorough clean/strip of the rest of the floor's residue and previous sealer.
Failing that, clean the loose out and fill with flexible grout and tone down with a sealer before sealing the rest of the floor.....it's a compromise though.