I'd present a juicy morsel of mathematics, say -- the Euler characteristics theorem for $\ \mathbb S^2.\ $ I'd adopt a classic proof.

Let $\ \mathbb S^2\ $ be divided into convex geodesic polyhedra, $\ P.\ $. (Allow some neighboring edges to extend one another so that occasionally they lie on the same large circle). Then the sum of the angles of a polyhedron $\ p\in P\ $ is equal to

$$ \pi\cdot(n_p-2)\ +\ A_p $$

where $\ n_p\ $ is the number of edges (or vertices) of $\ p,\ $ and $\ A_p\ $ is the area of $\ p.\ $ Then summing over $\ p\in P\ $ gives us the Euler formula rapidly:

$$ |V| - |E| + |P| = 2 $$

where $\ V\ E\ P\ $ are the sets of vertices, edges, and polyhedra of the given scheme.

The simple combinatorial argument must be satisfying to youngsters. On the other hand, the students get a feel for the place of the general theory since they would be pointed to the measure theory. Finally, they may appreciate the power of special examples, e.g. of surfaces of constant curvature. Indeed, one can go beyond $\ \mathbb S^2.\ $ One only needs surfaces for which the sum of areas of geodesic polyhedra would be, say, $\ -8\cdot\pi\ $ (instead of $\ +4\!\cdot\pi)\ $ and everything else would be the same.

>**From my personal experience:** I was invited to give a talk at a minor university (spring of 1996) where there was virtually no mathematics department and hardly any research to talk about. It was a relaxed 45-minute talk (in reality, under 40 minutes). Most of the audience were engineers (faculty and students; but the invitation came from an open-minded visiting experimental physicist).
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>I covered, no sweat: (0) Introduction; (I)Theorem 0 of the graph theory + *Königsberg bridges* Euler's theorem; (II) Euler characteristic for $\mathbb S^2;\ $ (III) Non-planarity of the Kuratowski graph $\ K_{3,3}.$
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>My audience was fine but nowhere as sharp or knowledgeable about mathematics as 9y old talented students.

I am willing to provide a detailed plan of the Euler characteristic portion of my talk together with the time schedule (the order and the details are important!) -- when this part is extracted and treated as a stay alone talk, it comfortably fits 15 minutes. During my lecture  I used but blackboard only. If you prepared some paraphernalia then it'd be even nicer. Do it yourself, don't let naysayers stop you.