Edmund himself initiates his process of repentance. True he does so only when having found himself in a much worse scenario than he had expected. He had assumed he would be welcomed by the Queen of Narnia a son, but instead was worked as a slave. Certainly this situational repentance does not fit Maimonides' definition of a full repentant: one who finds himself in the exact situation of when he sinned and this time chooses against sin. Nonetheless, that first opening is vital as it allows God to expand His presence in a person's heart, as explained in the midrash in the name of R' Yassa:
The Holy One Blessed be He said to Israel, my sons! Open to me in repentance even a doorway like the eye of a needle and I will open to you doorways that even wagons and carriages can pass through.
Summarizing, God promises that if one is willing to open their hearts to repentance even by the smallest of amounts, He will open their hearts the rest of the way. Edmund opened his heart slightly. First, he felt bad only for himself, but that emotion led him to differentiate between good (his siblings) and evil (the White Witch). Having identified evil, he began to sympathize, perhaps even empathize with the poor animals turned to stone by the White Witch. Finally, Aslan brings him the rest of the way, to become a true repentant.
Having achieved a full turnaround it is indeed not necessary for others to talk to Edmund about his past. This too is famously codified by Maimonides:
It is, nevertheless, a grievous sin to say to the penitent: "Remember your erstwhile conduct", or to remind them in his presence so as to put him to shame, or to mention things and subjects similar to them so as to remind him of what he did.
This also allows us to understand when Aslan himself brings up Edmund's past, urging Lucy to go on and heal others by rebuking, "Must more people die for Edmund?" Aslan himself does not hold a grudge having accepted Edmund's repentance completely. However, Edmund still owes an apology and must ask for forgiveness from each individual whose life was lost, or even made more difficult, due to his treacherous actions.
Can Edmund ever hope to achieve such forgiveness? How can one who directly harms hundreds or thousands of unnamed individuals find every single one of them and beg for absolution his sin? The answer, of course, is that he cannot. Yet, perhaps there is another way. One who has harmed a society can, achieve repentance by improving that society. Hence, Edmund becomes King Edmund the Just, "great in council and judgement." In this way, by contributing to society and to any part of society who approaches him, can Edmund achieve full repentance.
