On entering your Board constructor, assign this to a local variable (self) so other functions defined in the same scope can have access to it.
Something like the following will work:
var Board = function($element){
var self = this;
this.place_piece = function(piece){
this.board[piece.position[0]][piece.position[1]] = piece;
$element.drawImage({
source: piece.image,
draggable: true,
layer: true,
x: this.get_coordinates_by_cell(piece.position)[0],
y: this.get_coordinates_by_cell(piece.position)[1],
dragstop: function(layer){
console.log(self)
console.log($this)
console.log(layer)
}
});
}