what was ann`s decision?what favour did she ask kate in return

Ann is forced to show Mother the letter that Larry wrote to her before he died, which was essentially a suicide note. The note basically confirms Mother's belief that if Larry is dead, then Keller is responsible--not because Larry's plane had the defective parts, but because Larry killed himself in response to the family responsibility and shame due to the defective parts.

Mother begs Ann not to show the letter to her husband and son, but Ann does not comply. Chris returns and says that he is not going to send his father to jail, because that would accomplish nothing and his family practicality has finally overcome his idealism. He also says that he is going to leave and that Ann will not be going with him, because he fears that she will forever wordlessly ask him to turn his father in to the authorities.

Ann's decision, after witnessing Joe's confession about what really happened with Steve and the defective cylinder heads, decides to do nothing about Joe (not to turn him in, not to re-open the case and let Steve out after being proved innocent) and she asked Kate, in return, to tell Chris that she knows Larry is dead. That is, because Ann and Chris plan to get married, but Chris feels it's a little wrong - even though he loves Ann - because how can they get married if Kate disapproves? Kate, btw, disapprove of the marriage because Ann was Larry's girlfriend, but Larry is assumingly dead...