Does the impurity disappear from a washing basin in which soap residue has frozen and which I was unable to remove when I sprayed water into it, and if it does not disappear, what is the solution?

There is no obligation to remove traces of congealed soap. Rather, what is obligatory is to remove the the impurity itself and its attributes if they are apparent. If traces of the impurity are not apparent, then it is sufficient to run water over it. The mere flowing of water is sufficient to remove what is not the impurity itself. As for the impurity itself, its attributes must be removed, and they may be removed by merely running water over it.