where and how, in the course of the 'letter', does king attempt to make common cause with them?
king addressed the "letter from birmingham jail" to eight fellow clergymen who had written a statement criticizing his activities.
King's attempts to associate himself with his critics begins with the salutation of the Letter--"My Dear Fellow Clergymen"--which helps establish two important elements: 1) King is reminding the clergymen who have criticized his actions that he is one of them; 2) He begins the letter with a tone of conciliation, consistent with his adherence to non-violence in all actions, including responding to criticism. Also in the opening paragraph, King says that their criticisms require an answer because "you are men of genuine good will" and the criticisms are "sincerely put forth." From a rhetorical perspective, King has quickly established a tone of reasonableness that he carries through the entire letter, even when he has some relatively harsh comments to make about such people as "the white moderate" and his disappointment with the church.
Later in the essay, he is careful to note that their criticism about "our willingness to break laws" is entirely legitimate, which acknowledges not only that their concern is a reasonable one given the circumstances but also allows King to launch into an extended and very logical analysis of just and unjust laws. During this discussion, King says, "Let us consider a more concrete example of just and unjust laws," a very subtle, but effective, technique to remind his critics that he is talking to colleagues, not adversaries.
Even when King discusses his disappointment with the "white moderate" and the church's response to King's attempts to right the wrongs of racial injustice, he remains reasonable and considerate when he says that if anything he's said "overstates the truth and indicates an unreasonable impatience." His long discussion of his disappointment with the church is meant to stir in his critics some embarrassment, but he doesn't make this attack in a verbally violent way--his argument is logical, reasonable, and one senses his own feelings of quiet disappointment. In an important gesture of good will, King asks for their forgiveness if he's overstated his case, again subtly reminding his critics that his letter is written with the same respect and sincerity as their original criticism.