Unfortunately, my prompt requires me to focus on Cordelia's death only, so I won't be able to venture into other elements of the unjust in my paper.
If, in answering the prompt, you want to say that Cordelia's death contributes to the theme of an unjust world, then you ought to marshal other evidence of this theme in the play, briefly listing other evidence in order to show that this is, indeed, among Shakespeare's themes of the play.
Similarly, when you say "there is a sense of redemption in all of the other deaths in the play, but none in Cordelia's death" then, obviously, you will have to talk about the other deaths in order to make that point.
In order to answer the questions "Why does the death of Cordelia in King Lear matter?" and "What was Shakespeare's purpose in Cordelia's death?" you must talk about other aspects of the play. In order to talk about Cordelia's death in the context of the play, you must talk about the rest of the play, or at least other elements that support whatever it is that you are arguing about Cordelia's death.