Simply use a mild soap and sponge and gently work on small areas at a time.
You will need to empty your water and refill regularly if your fireplace has years of dirt.
Keep a tea towel or old T-shirt handy to wipe up drips and and mess.
I used a standard kitchen sponge with a slightly more abrasive side and a soft side. I needed both sides of the sponge as this fireplace had not been cleaned properly since 1970 or earlier.
DO NOT USE : lemon juice, vinegar, bleach or any caustic products as these will damage the limestone
You can see the comparison between the grey blotchy discolored limestone and the bright white cleaned stone.
This fireplace was hand carved in the 1950s in Carcassonne by local masons. The motif is the castle of La Cité.
Almost finished the cleaning process now ...
The original architects drawing for this fireplace in 1958
The original receipt from André Pitié of Castres 1958