Advice on teaching 8 year olds king and queen vs king?

A more advanced exercise  would be to do it without moving the King at all in certain key positions (like this one) :)

Showed this to a few kids at the club and they have a lot of fun trying to figure this out ... and it teaches them even more valuable lessons about how to use a Queen.

Update: One solution is provided here.  

Shivsky = Chess troll?

Apologies..should have clarified. It is "White" who cannot move his King, i.e. only uses his Queen. If Black cannot move his King, I really am quite the troll :)

That's what I taught my brother, and he got that pretty quickly. After that, the K+R and K endgame was easy.

With King + Queen against lone King lessons, I start with bringing the Queen to a position on the board that boxes in the opposing King as the first move. I tell my students to keep the Queen a Knight's move away from the opposing King. This keeps the student from moving their Queen onto a square where she'll be captured. After the opposing King has been limited to the box, we start moving our King closer to the opposing King. This would be a good time to introduce the idea of "King Opposition." I refrain from making the box around the opposing King too small (with additional Queen moves) until our King is closer. Once the King gets closer, the Queen starts to limit the opposing King's moves. As the box gets smaller, I bring up the topic of stalemate. I usually use a mating example that gets the Kings in opposition with one another with the opposing King on the back rank or side of the board. For their first introduction to this checkmate, I use an example that is very straight forward and visually easy to follow (later on we look at a few different ways to use the King and Queen to mate). The last move will be the Queen delivering the mate. Stalemate is the biggest problem you have when teaching this mating technique to kids.