Why does Jake choose to take up and continue the defense of Carl Lee?
This is a great question. There are several reasons why Jake takes up the case and decides to defend Carl Lee.
First, we must admit that Jake is not doing it all for the sake of altruistic motives. For example, Jake knows that if he wins this case, he would win great renown and that he would be considered one of the great lawyers in Mississippi. This is an important motivation and his showmanship in the novel points to this motivation at times.
Second, Jake sees the injustices; he see the evils of the KKK and the oppression of the the black community. As a lawyer, he also wants to do the right thing. In this case, it is to defend Carl Lee.
Finally, he feels empathy. Part of the reason for this is on account of the fact that he has a daughter as well.