You will need an item to light up the tiles first, then push the blocks.

There is a hint to the item underwater in the SE of the SE-Penitent Lake. (Right next to where you would use the item first).

The item is buried in the SW of the SE-Penitent Lake.

Dug everywhere in the SW but no joy? Do you mean the area beside the half ladder? I managed to jump on that ladder but I'm blocked by a gate.

You dug everywhere you could in the SW section of the Penitent Lake map but did not find a buried chest with a blue gem? It was in a snowy area on the ground floor near where you fought a rock monster. This is in the SW section of the Penitent Lake map but has nothing to do with the gate in the SW section of Penitent Lake. (I am still puzzled by that gate myself). If you look up you should see a golden serpent statue nearby (the statue is not related to the dig. It is just another landmark).