Titles with two meanings, y'all: that double entendre magic is what makes titles like Adaptation and Melancholia and, yes, The King's Speech so punchy. This title has multiple layers, like the best cakes:

Layer #1: we know that the climax of this movie is the wartime speech that Bertie needs to give as King of England. So in that sense, it makes sense to name the movie after the speech he has to give.

But (and here comes layer #2) in a broader sense, the movie is about the king's "speech," meaning the way the king talks in general. The movie as a whole is about his speech impediment, which he endlessly struggles to overcome. So when you look at it from both angles, the title was sure to be a shoe-in as soon as it first came to the producers' minds. There's just no way anyone was going to improve upon it.