My first thought is a talk titled something like "What we can't do, infinity, and even bigger". It would cover two proofs that I think have a particularly high coolness to complicatedness ratio: Cantor's proof of the uncountability of reals, and Turing's proof of the undecidability of the halting problem. The outline would perhaps be something like:

1) Infinities are not all equal. There are lots of integers, but there are WAY more real numbers. Spend 20 minutes explaining diagonalization, and what it means.

2) Computer programs can do a lot, but not everything.

Spend 20 minutes explaining the halting problem, and what it means to be undecidable. I'd probably avoid defining Turing Machines, since that seems like a mess, and just talk about it in terms of computer programs, which are hopefully a little more familiar

3) The punch line: Computer programs are countable, like the integers, but the problems that you might want to solve (languages) are uncountable, like the reals. So although computer programs can solve lots of things, actually, most things can't be solved. Hopefully this can be done in 10 minutes given that I've already talked about uncountability and undecidability.

I like these proofs since they don't require any machinery, and seem like they can be done visually with some animations. Thoughts?