There is grout paint after you have cleaned, but can't get it any cleaner. Google "grout paint" and you will find some info on it. I have some but haven't used it yet:) I would use a light to medium gray.

I would start with soapy water (maybe Dawn) and a scrub brush for cleaning.

Thanks. I forgot to mention that I tried cleaning a small section with Aqua Mix Nanoscrub from Home Depot. It did not alter the appearance at all, unfortunately.....

Once you are sure there are no lose particles and you have done your best to clean, try the grout paint.

https://www.younghouselove.com/2014/01/how-you-like-me-now-grout/

I am not sure i would use as light a color as she did:

Sadly, many people over the years put self polishing wax on slate....which builds up like bad varnish over the years. I know you said non toxic...but the best thing to remove it with is ammonia, or a cleaner with ammonia, and a scrub brush. And clorox will often bleach out soiled grout. There is a Clorox Gel grout cleaner that will stay in place until you wash it off. Then seal the grout with a sealer. I find grout paint looks like painted grout....which I think is un natural.

After you scrub it, try steam cleaning. But try it on an unobtrusive area first, and do your proper researching, since Ive never had slate.

To clean grout, you could try powdered Bar Keeper's Friend cleaner.

We had such tile in our old house. I regularly mopped with a product from the stone/tile products section at Home Depot. Maybe TileLab? It was labelled as a stone cleaner, maybe a green bottle. Sorry I can't recall the name but it is not with the cleaning product section. I diluted it, and that floor looked great. 1100 sq ft in all, and I installed every inch of it myself.

It almost looks like the tile sits above the grout in places , could be just the angle. As for cleaning all the above are good ideas but don’t expect a change it the actual slate. I would go with a very dark gray grout paint if you go that route.

A professional with the right equipment and cleaning chemicals is in order. You can't get that wax off without that. It's a 2 step process. Chances are you can't clean the stone and the grout in same step. It basically needs to be re-finished. An on site consult from a marble and stone restoration company can provide you with the actual steps and cost associated. Oh, and don't use grout stain (it will soak into stone, doesn't look authentic, won't last, and will cause greater challenge long term if you want to properly restore it). That's our recommendation. Good luck.

Thanks everyone for the suggestions. This is really helpful! Really appreciate it!

You can pick up a grout cleaning brush at Home Depot, they are great, skinny little brushes with very stiff bristles. I used Simple Green and mixed bleach with it, made my grout look like new again. It did take a few old towels for wiping with after I sprayed and scrubbed. I too second the idea of painting the grout dark once you've done what you can.

Oxy-Clean powder mixed with water and a stiff brush.

The light colored-grout in jhmarie's example looks fantastic!! I'd go that route, using the grout paint. You could even try a patch of light vs. dark and see which you prefer.

I would suggest steam cleaning as well. Might be best to consult a professional, but I've used a hand steamer and cleaned white grout lines with porcelain tile before. It made a huge difference. I steam cleaned and let it dry for a few days (with no foot traffic to get it dirty) and then re-sealed the grout.

I have 2 rows of black slate tiles in front of my fireplace that were spotted and faded. Tried various methods of cleaning and restoration to no avail. Finally tried black shoe polish, the kind that comes in the little flat tin. Rubbed it into the tile, avoiding the grout. It was like a miracle! Instant restoration. This was over 5 years ago and the tile has looked great ever since. I wouldn't try it on grout, though.