He pushed Bran out to his paralyzation, not his death. It was not murder, it was attempted murder. It was a heinous act. Can Jaime atone for it? Mayhaps Bran could forgive Jaime, but the act cannot be undone. I'm glad that we act with continued scorn and vehemence towards Jaime. He is a vile man. His longing for redemption is good, but sins are legion.
I find it interesting that the words of the High Septon (spoken by Pycelle in the show) at Eddard's execution were, "As we sin, so do we suffer." In Westeros, we know that is a lie. So, Jaime, Walder, Roose, and Ramsay may never get what is coming to them.
Discussing the attempted murder of a child, mayhaps we can put Jaime, Sandor, Gregor, Amory Lorch, King Robert, Darkstar and the violent men of Westeros into some perspective... Two nights ago and just eight miles from my home, a six year old girl was murdered. What can the murderer do to prove to me that he deserves to live? Nothing. Absolutely nothing. He also killed eleven other people that night in Aurora, Colorado. Twelve dead. At least fifty-eight wounded. It also appears that he booby-trapped his apartment, cranked his stero, and left his door unlocked when he left in order to entice neighbors or the police inside in order to murder more people. Who will forgive him? His living victims? The victims' families? The community? I think forgiveness will be in short supply for this murderer. Will he attempt to atone for the lives he took and the lives he's shattered? At this point, it seems unlikely.
Eddard's refusal to support the murders of Rhaenys and Aegon, his hiding of Jon, and his resignation rather than back Robert's scheme to murder Dany and her unborn child marks him as a man of character. As Eddard sinned, so did he suffer... What a lie.
In a brutal world, Sandor understood better than anyone. A knight is a sword.
I'd love to see the Lannisters, the Freys, the Boltons, Randyll Tarly, the Greyjoys, Gerold Dayne and all the other villains get theirs... but I'm not holding my breath.