Get rat in leprechauns cave to leave.
+2 open cupboard, +2 for retrieval, +2 for giving to rat.
In order to get the cheese, you must first make it into the Witch's Gingerbread house. The cheese is in the cupboard (circled in image to left).
The Witch may come home at anytime while you are in the house. Either move quickly to retrieve the cheese or push her into the cauldron first.
The big rat guarding the door in the leprechauns cave likes cheese (don't all rats?), and he'll gladly take it and disappear. This is the best way for dealing with the rat, but be careful not to get too close to him or he'll shred you!