Vocabulary in A Scandal in Bohemia

A brougham (pronounced "broom" or "brohm") is a small, enclosed carriage designed for traveling quickly, especially in cities.

Bohemian, in the late 19th and early 20th centuries, referred to those who followed an unconventional lifestyle.  Anyone who did not follow a socially accepted path in life might be characterized as bohemian. Holmes qualifies because he follows a new and largely unknown profession.

Here, "exalted" means a noble, elevated, or lofty nature. The word underscores the status of the client, the King.