I think we all agree that the survival of this state aren't very likely, but there's nothing ASB over its survival, it simply demand that it develop some ways to survive, just as the Normans did. The survival of Normandy was also a near thing, but the Norman called on Danish allies in the early reign of Richard I. So let's say that the Berbers try to throw them into the sea but the king call on Norse allies. Harald Bluetooth again wouldn't be a bad choice, some of their Norse allies are rewarded with land and they bring Christianity with them. This create a minority among the local Norse who are Christians, which push the king toward converting to Christianity.
I'm not sure what survival strategy the Maghreb Norse develop, but we can be sure that, while they're pagan, they're likely raiders and pirates of both Christians and Muslims. Their conversion to Christianity likely lead to this being slowly limited except against Muslims. So I expect that they will own Sicily, Sardinia, Corsica, Malta and Balearic islands from their pagan period. I also expect them to push some kind of feudal model on their domains. To deal with the Berbers they will have to have excellent light cavalry, likely mixed with strongly fortified coastal cities, some kind of manor system etc.
Their domains speak a mix of Vulgar Romance, Berber and Semitic with a few Greeks. The Norse will have no wish to integrate into the local culture, like the Normans did. So I expect early on the Norse language will do well. They're also much better able to keep contact with the homeland. Of course down the road they're a small minority. But on the other hand their subjects are a mix of different people. If Norse survive, it will likely do so in a relative limited areas (I could see Malta as a stronghold of the language). Even if it survives spread out all over their empire in small enclaves, the elite have a high chance of adopt some Italian dialect with heavy Norse influence. The Romance and Semitic dialects of the empire will likely be heavily influenced by Norse, but the Semitic will likely also see a large effect from Romance and Berber. Berber will likely be less influenced, but it may also get less influence from Arab.
I expect as the Norse convert, they will begin a slow conversion of their subjects, I expect with the Norse conversion having begun around 1000, there will likely be no more Muslims left by 1300 outside a small minority, most will have converted, while the last stubborn one will have been expelled.
A interesting factor are that Morocco and Muslim Spain will be isolated from the rest of the Muslim world. Linguistic I imagine that one result will be that Morocco will be Berber speaking, with only a small Arab speaking minority. While Morocco will stay Muslim unless conquered, I also expect that Morocco will be much more European in culture, as they're isolated from the rest of the Muslim world and have to become a integrated part of the European political scene to have Christian allies against their Christian enemies.
The West African slave trade will likely also suffer, the medieval Europeans doesn't have the same demand for African slaves as the Arabs did. This will also limit the spread of Islam in Sahel. Of course with a less developed slave trade, the Europeans will lack slave trade infrastructures, when they set up the Transatlantic Slave Trade. This is interesting in itself. We may also see better state structure developing in West Africa. If this fall together with the spread of Catholicism in the region in the Mediaval Age. It may also result in the area being much better integrated into Europe with the benefit that gives. Maybe early modern western Africa as a kind of African Russia, somewhat undeveloped, backward compared to early modern western Europe, but seen as part of a greater Christian/"European" civilisation?
) earlier, causing the Pope to call in the Normans to take out Sicily, and they do well enough that they keep going on a papal-sanctioned rampage across the Mediterranean.