Since you deal with higher ordinals, 10 minutes seems like more than enough time to prove that there are more real numbers than natural numbers. I think most people can follow the "list of decimals" proof. After you do this, your audience is ready to believe that there are different infinities. Now, exlain that YOU are interested in the problem characterizing these infinities.
You want to show them that there are limitations on proving the existence of higher ordinals. Of course, you cannot do this. But, for example, you could use Russell's paradox to show that there are subtleties that one must take into account.