Slate in a shower / bath area can be a constant problem to clean. The picture shows the result of cleaning with a stone safe soap film remover. However the soap scum was there long enough to etch into this soft stone.
To help prevent greater problems with slate in showers, I would suggest regular sealing with a professional stone care product. As often as every 6 months after some serious cleaning with a stone safe soap film remover. This will make it easier to maintain.
The Cleaning Superhero can advise you about the best course to take when cleaning a slate shower or bath area.