You will find so many examples of each in the letter that you will be in the enviable position of choosing which you like best. In terms of a simple sentence, I like where King clearly states that he and his followers know what they are doing:
This reveals that we did not move irresponsibly into direct action.
This is a direct sentence that Dr. King uses to articulate the condition of the movement and the need for it to become a reality. It’s a simple sentence and it is used to emphasize the need to take action in a concise manner.
For a complex sentence, I like a moment where Dr. King calls Socrates to assist him in his cause:
Just as Socrates felt that it was necessary to create a tension in the mind so that individuals could rise from the bondage of myths and half- truths to the unfettered realm of creative analysis and objective appraisal, we must see the need of having nonviolent gadflies to create the kind of tension in society that will help men to rise from the dark depths of prejudice and racism to the majestic heights of understanding and brotherhood.
This is a great sentence that is complex in its construction and in its ideas. There are multiple phrases combined to create a transcendent effect in the reader. As I said, the letter is filled with examples of both types of sentence and it will turn out to be your choice for selection.
Martin Luther King, Jr.'s "Letter From a Birmingham Jail" was written after King had been arrested in April of 1963.
Dr. King was an extraordinary orator; his writing is moving, and sophisticated in its structure and organization. Finding a simple sentence in this letter was not the easiest thing to do, though—ironically—it is the simplest kind of a sentence.
The simple sentence is:
...a single independent clause with no dependent clauses.
At one point Dr. King writes the following sentence referring to "nonviolent direct action"—
It seeks to so dramatize the issue that it can no longer be ignored.
This is an independent clause (a complete thought—a sentence that can stand alone and make sense) with no dependent clause (and incomplete thought) included. Another form of of sentence is the compound sentence. A compound sentence is like the simple sentence in that it uses the independent clause, but this time is has multiple independent clauses (exactly two, in this case)—and still no dependent clauses are used.
These clauses are joined together using conjunctions, punctuation or both.
Dr. King, in the introduction to his thoughts from the Birmingham jail, makes the following note. This is a compound sentence: two independent clauses joined by a conjunction.
We have some eighty-five affiliated organizations across the South, and one of them is the Alabama Christian Movement for Human Rights.
A complex sentence is a sentence with one independent clause and one dependent clause. The dependent clause cannot stand alone, therefore, it "depends" on the independent clause (that can stand alone) to make sense. The following is a complex statement. The portion of the sentence that begins with "We" and continues on to the comma after "mutuality" is the independent clause. It is a complete thought—it can stand alone and make sense. The remainder of the sentence following the comma is the dependent clause that makes no sense alone, and so depends on the independent clause.
We are caught in an inescapable network of mutuality, tied in a single garment of destiny.