Welcome to the forum, Michelle!

Sorry for the delay in a response.
I think the cat litter itself has a wide range of pH. If it happens to be on the acid side of "pH Neutral", it may be damaging the slate by etching it. A picture may help us identify that. If it is etched, no cleaner can do anything to improve that type of damage.
But absent that, I'd perform a test.
Start with a small test area the size of your hand. Start with a stone-specific pH neutral cleaner (you'll generally find this in the tile aisle at the box store or at a real tile store...but don't expect any real info out of the salespeople, as only 0.1% have ever even heard or used a 'stone specific cleaner'). Apply the cleaner and allow to dwell for at least 5 minutes. Then agitate with a fine nylon bristle brush. The finer the bristles, the better. Then wet vac the mess away so that you don't re-deposit the mess to another part of the tile, as is so common with mopping or cleaning with a rag. Allow that to dry to evaluate your results. Come on back and let us know what's happening.