Edward decides to perform the menial tasks because of the example of Alfred the Great, "another English king (who) had a commision like to (the one presented to Edward), in a bygone time". Edward figures that "it is nothing against (his) dignity to undertake an office which the great Alfred stooped to assume", and so decides to try to complete the chores the peasant woman assigns him.
Having made this decision, Edward is first told to help with the cooking, but, being unused to such work, Edward, sadly, like Alfred before him, lets the cookery burn. Fortunately, the woman returns soon enough so that the entire meal is not ruined, and in the end, Edward manages to make "a hearty and satisfying meal, and (is) greatly refreshed and gladdened by it". When breakfast is over, the woman puts the little king to the task of washing the dishes; this command is "a staggerer for a moment", but Edward, remembering the example of Alfred the Great once again, undertakes to try it, although he makes "a sufficiently poor job of it". He then is asked to help the little girls pare some apples, sharpen the butcher knife, and card some wool, at which point Edward has just about decided that he has emulated King Alfred "in the matter of showy menial heroisms that would read picturesquely in storybooks and histories" quite long enough. Edward's final assignment is to take a basket of kittens and drown them, but he avoids having to do this onerous task by fleeing the house, having just discovered that the "rascals" who have been pursuing him have arrived.
In addition to Edward's desire to emulate Alfred the Great, I think it must be considered the extent to which he decided to perform the tasks assigned him because of the generosity and kindness of the peasant family in accepting him into their home. Edward had been wandering, completely penniless, friendless, and either threatened or denied by almost everyone, when he happened to stumble across the peasant family's welcoming abode (Chapter 19).