How can I repair cracked stones around a drain in a wet room?
That looks just plain done wrong to me. The drain entrance should be below the membrane that should be below the stone. Cracks in the grout should have no impact on the waterproofing effectiveness. Drain entrance level with stone surface = designed to fail.

A minimally invasive fix would be to lay a membrane on the top of the current stone and lay new stone above that. Making the connection to the side wall membrane will be somewhat tricky - assuming that's even there, which I have to wonder about... 

If you want to save the current stone the options are purely invasive/destructive; and that's if you can get the stone up without destroying it.