There's some nuance I'm missing here. I'm assuming you can't walk along the path straight - fair enough. But usually for ice it goes with a system of 'careen until you hit the wall', and that looks impossible. Can you move the rocks, maybe?
I'm pretty sure the ice cracks behind you and you have to walk on every tile on every floor before being able to access stairs to the next floor, or something like this. Basically step on every tile on the floor exactly once before you can progress type of puzzle.