The two main places where I see redemption in King Lear are in the stories of King Lear and Gloucester, respectively.
Both Lear and Gloucester sin, if you will, early in the play. Lear is cruel to the daughter who least deserves it, while Gloucester disinherits Edgar.
Both Gloucester and Lear suffer terribly as a result of the wrongs they have done to their children. Through their suffering they come, by the end of the play to have more empathy and humility and indeed humanity than they had at the beginning. This, to me, is redemption.
King Lear is perhaps the most disturbing and the most ruthless tragedy of Shakespeare. Redemption in the world of the play at least apparently is rather repressed if not absent. On the level of paternal relations and emotional economy, there is misunderstanding ranging to incomprehension. Then, there is betrayal both fraternal and paternal; there is political betrayal too in the world of the play. As the mad Lear says in the Heath scene, it is indeed a case of the 'unaccommodated man'. It is, as it seems not even an indifferent universe but rather hostile and sadistic.
But deep within, there is a trace of redemption in the retributive justice that leads to the death of all the conspirators. Edgar's destiny and the way it has an upward swing at the end of the play is definitively redemptive. His values are also noble. His haunting line that states it is never the worst as long as one can say this is the worst underscores the fighting resolve and the stoical dignity of man that is also apparent in the final state of Lear. His madness is not just a downward move but also an overreaching of the state of commonplace rationality to glimpse that which lies at the beyond of reason. Cordelia and Lear reunite in the final moment, and as Freud would see it in a mother-son dyadic form, in their death. Lear's 'look look' is his cryptic vision pointing towards an alternative redemption at the end.
The theme of 'redemption' is not stated explicitly by Shakespeare in 'King Lear' because Shakespeare likes his audience to think about the characters who achieve redemption on a "man's (or woman's) deliverance from sin and damnation basis". There's the prospect that those who are 'pure evil' such as Goneril, Regan, Cornwall and Oswald have no opportunity for redemption but those that possess various shades of 'good' such as Edmund, Lear, Cordelia, Edgar and Gloucester at least redeem themselves on a personal and 'each other's eyes' basis even though the tragedy is that the wisdom of their errors and follies comes too late.
Therefore, the theme of redemption functions on a personal, familial/domestic, public, theosophical/philosophical basis to show the effect of individual's 'sins' on self, family and nation. Despite the chaos that ensues in Act 5, with so many dead, the country is spared the rule of evil doers and from the ashes of chaos an era of good is ushered in by new king Edgar who will see harmony and prosperity reign in 'old Albion'.
To add to the above ideas, also another portrayal of regeneration in King Lear is that by the end of the play, there is nothing standing in Edgar's way to re-generate society and move it forward due to the events of the play.