In the Game of Thrones episode "The Long Night" did Jaime have any impact on the course of the battle or the main characters? Brienne defended him in front of Daenerys and the Starks in the episode prior and Bran acknowledges their last meeting and let's him off the hook but for what purpose? Jaime has done nothing to prevent the battle of King's Landing or fulfilled any action that seems worthy of being forgiven for all his prior misdemeanours.
Whilst amflare's answer gets to the heart of it:
- He had Widow's Wail a Valyrian steel sword that is top notch for fighting White Walkers and their wights.
- He is a good fighter, one of the most renowned in all of the Seven Kingdoms, a knight and a former member of the Kingsguard. Even without his right hand he can still give others a run for their money.
- They needed every body possible, there and one as good as Jaime is too good of an opportunity to miss.
In the scale of the battle it's safe to say that he would have chopped down a large amount of wights defending Winterfell.
However, there's another main point, Jaime is a good battle commander and leader. Whilst Brienne was in immediate charge of the flank held by the Arryns, Jaime is a commanding presence. He's been shown to be a strong leader and a half decent battle commander, he's the sort you'd want as a second in command on the field to help organise the troops. In fact if I'm remembering correctly I think he does give orders or relays orders down the lines in the chaos which is very useful in the heat of the conflict.
It can be argued that no one had an impact on that battle aside from Arya. The Night King was rolling them all up pretty neatly.
As for Jamie's individual contribution though, he was a reasonably competent fighter with a Valyrian steel sword, something that is quite helpful when fighting White Walkers.
Daenerys and the Starks let him off the hook because they needed every living body they could possibly get to assist in the fight.