How does king balance the twin appeals to religion and patriotism throughout "letter from birmingham jail"? do you think he puts more emphasis on religion or patriotism? why do you think he makes that choice?

King balanced religion and patriotism by placing biblical calls for justice side by side with the need for justice in regard to civil rights in America.  He doesn't so much emphasize one over the other, but rather makes the point that if we truly act on our religious convictions, we will apply those convictions to matters of justice in our relationships with our fellow citizens.

Some examples of how Martin Luther King, Jr., applied assertions about justice from the Bible and church fathers to America's need for justice in regard to the civil rights movement:
... He said he was had come to Birmingham because injustice existed there, in the same way that Old Testament prophets had left their own towns and villages to go and confront injustice in other towns in their day.
... He quoted Saint Augustine, who said that "an unjust law is no law at all," to support his point that we have a moral obligation to disobey unjust laws.
... He pointed to how the early Christians were "disturbers of the peace" because they challenged the existing morals of a morally bankrupt society, and called upon Christians today to act with the same sort of conviction.

The entire 1963 letter from the Birmingham jail was addressed by King to his fellow clergymen--particularly white clergymen--who kept saying the civil rights activists should settle down and be patient.  King brought forth powerful religious arguments to say that men of Christian conviction could not stand by idly while injustice continued to be perpetrated.