Mindfulness can be of assistance with the problem of stammering. The British Stammering Association has on its website an interesting article about a course for stammering based on Mindfulness principles and approaches being run at London's City Lit. Here is the link to the article.
In Zen Buddhism there is the story of the Zen master who says to his pupil, "One must never think of the white monkey," if you want enlightenment. You can guess what happens. Thinking about not thinking about the white monkey is the same as thinking about the white monkey. Thus, thinking about not stammering is the same as thinking about stammering. "This" and "not this" are the same thing.
Here is the King's radio addess of 3 September 1939 on the outbreak of World War II which is read by Firth in the film. A brief portion of the King’s final annual Christmas address, in 1951, which was recorded in sections and then edited together (due to the King’s poor lung capacity as a result of lung cancer and the removal of his left lung), can be found here.
Here is another article about Mindfulness and stammering which may be of interest.
Finally - This blog sets out a simple form of mindfulness sitting meditation.