He is using his credibility and trying to show that he is a good person. He is saying, or asking, himself if he can accept blows and attacks without retaliating, and the answer is yes. He is telling in the letter that he is a good person and even though he is doing nothing wrong and people are treating him wrong, he knows the consequences and he is willing to accept them without fighting back.
Ethos
We began a series of workshops on nonviolence, and we repeatedly asked ourselves:'Are you able to accept blows without retaliating?''Are you able to endure the ordeal of jail?'”
He is using a lot of emotion words and he is trying to convey an emotion of guilt and sadness for all of the people that had to suffer because of this violence. He is trying to get to people's emotions and make them feel guilt for what they have done, not to make them feel bad, but to motivate them for change.
Pathos
But when you have seen vicious mobs lynch your mothers and fathers at will and drown your sisters and brothers at whim; when you have seen hate filled policemen curse, kick and even kill your black brothers and sisters; when you see the vast majority of your twenty million Negro brothers smothering in an airtight cage of poverty in the midst of an affluent society;
Martin Luther King Jr. is trying to use logic to explain why they did not need to use violence when he was being peaceful. He is using allusion as well to known people to show that he was was doing nothing wrong just like them so if they were to give him a punishment it would be the same as giving those people a punishment. He does this so people understand his point of view and are motived to change.
Logos
Isn't this like condemning a robbed man because his possession of money precipitated the evil act of robbery?Isn't this like condemning Socrates because his unswerving commitment to truth and his philosophical inquiries precipitated the act by the misguided populace in which they made him drink hemlock?Isn't this like condemning Jesus because his unique God consciousness and never-ceasing devotion to God's will precipitated the evil act of crucifixion?