Now you can probably give the kid a hint, like "don't you think it might be better to not advance the pawn? It does seem to be getting in the way, right?" to speed things along. The next try will go something like this:
The kid is clearly making progress. He has understood that the pawn must not be advanced too early. He got his king up to the 4th rank. The problem was that he did not know how to push the black king back. He did not know the concepts of 'opposition' and 'outflanking'. That is why he made the move 3.Kd4, when an experienced player would play 3.Ke4, taking the opposition.
So now you switch games. You take the pawn off the board, leaving just kings facing each other. Explain to the kid that the goal is to get the king to the other side of the board. White moves first (kid has white):
Soon the kid will see that it is impossible. "That's not possible," he says. "Right," you say, "because Black can always block the king. This is called the opposition. Let's try it again, but this time Black moves first."
Now reset the kings on e5 and e7, and Black moves. You move the king to one side (e.g. d7) and he will see that his king can quickly reach the back rank by playing Kf6, then Kg7 and Kg8. Explain that this is called "outflanking". So, if you have the opposition, you can either block your opponent (defense) or outflank him (attack).
Now return to the original endgame with king and pawn against king. It should take one or two more tries for him to get it right:
If the kid made some incorrect moves which slowed down the winning process, or if he seems unsure, let him practice it again until he has got the pattern down.
If there is still time, you can let the kid play the defensive side from a drawn position (i.e. after the first move 1.e4). Make sure he understands that the black king must step straight back when forced to retreat. Another thing to teach would be the position with White to move, the white king on e4, the black king on e6, and the white pawn on e2. Here White does not have the opposition, but by making the "tempo move" 1.e3, he gets the opposition and the black king is forced to give way.
I have taught this lesson to many kids and have not yet found a kid who could not eventually solve it with trial and error. Usually it should take about 20 minutes. Of course, every kid is different, so the above examples of play are approximations. But the general idea is to let them play, you play the defense, and use the king vs. king game to teach opposition.
You can also teach this lesson to a classroom, taking suggestions from various kids, or by picking volunteers to take turns trying to win for white. It usually makes for a fun lesson for the kids! Ciao!