Alright, so I started a new campaign with Vestfold instead, and I unified Norway.
What it does: a few event screens that don't do anything, all settlements and armies of the enemy jarl become yours, your faction sign changes (the thing you see on the shield in the interface), maybe the banner changes as well I don't know.
So, I don't know how to change your faction sign, and the enemy's lands and armies are probably irrelevant since you already destroyed him.
What you can do is to give your leader the "crown of Norway" (it gives a few cool bonuses) via the console with: give_ancillary "Your Faction Leader Name" "Norway_King" 1
This crown ancillary doesn't show up even when you unite Norway properly ingame, but it's present in the files, so I don't know if its a bug or if the ancillary comes from vanilla. But in any case, it's a nice reward for your efforts
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You can also add the trait "King of Norway" with the console typing: give_trait "Your Faction Leader Name" "NorwayHero" 1
And if you are playing with Vestfold, you can change Harald Tangledhair into Harald Fairhair (to follow the saga, he didn't cut his hairs before he became king of Norway) by typing: give_trait "Harald" "HARALD" 3. Then again, I don't understand why that didn't happen ingame, this trait clearly seems to come from this mod. Anyway ...