When it comes to Sweden, it's less a question of when and more of by whom. Sweden was established as a medieval christian kingdom by one man. And that man was named Birger Magnusson (ca. 1210-1266). Birger was the Jarl of Sweden under king Erik Eriksson (known to history as Erik Lisp and Lame). He was a weak king, as his epithet atests, and was dominated by his Jarls, first Ulf Fase and then Birger. Birger then married the kings sister, and upon Erik's death had his son Valdemar elected king (Scandinavians counted dynasty matrilineally as well as patrilineally).

The Jarl (capital J) was basically the shogun of Sweden. He led the army and was theoretically the number two guy under the king. But if the king was young and/or weak - his rule was absolute. Most Swedish kings before Birger's son Magnus Birgersson ("Ladulås") were very weak.

Institutions

First, the most important change of the transformation is the one you didn't mention: Institutions. Specifically there is one such occurrence that is vital for the formation of a cohesive state: Taxation. Sweden did not have taxation until 1247, and the Battle of Sparrsätra. The battle was a result of Swedish resistance against having taxation enforced; They lost. The church annals thus state that the Swedes lost their freedom that year, and had several taxes imposed upon them. Note that the Germanic peoples equated taxation with slavery - it was something you imposed upon defeated nations.

Military

As stated, Sweden had no taxation before 1247, it instead had a navy. The ledungen system that had existed before entailed that all men had to arm themselves and serve in the navy when called upon. Each "härad" (hundred) was to supply one longship and crew for the king's campaigns. It was through this system of military service that the first taxation was created. The compulsory military service was redefined into a standing tax that was paid to the crown. An excellent example of the levy-fleet in action is the portrayal of the second Finnish crusade in The Chronicle of Duke Erik, which I'll try to track down an English translation to quote from. Sweden then created a new military system consisting of European heavy knights. The "Ordinance of Alsnö* from 1280, under the rule of Magnus Birgersson ("Ladulås"), states that every man who served the king as knights were exempt from all taxation.

On Swedes and Sweden

All people of Sweden are not Swedes, there is a clear divide between Swedes and Geats, and it exists to this day a clear dividing line in Swedish accents between the two. The Geats of the south were Christianized much earlier and were also subject to taxation. That is where the two Swedish royal dynasties had their strongest support: The Sverker dynasty in Östergötland and the Erik dynasty in Västergötland. (Both are named so because of their liking of the names Erik and Sverker respectively). The Swedish royal castle at this time was Visingsö, in the middle of lake Vättern that separates the two provinces. Their control over the Swedes around lake Mälaren was light, and they were dependent on the local nobility. But even then the Mälaren region had a higher population, it was just that the kings didn't gain anything from that. Birger Jarl made it the center of the Swedish kingdom and founded Stockholm.

Sources: Harrison, Dick : Jarlens Sekel. (The Century of the Jarl)

Lindström, Henrik and Fredrik: Svitjods Undergång och Sveriges Födelse. (The fall of Svitjod and the Birth of Sweden)

Erikskrönikan (The Chronicle of Duke Erik)

Can you elaborate on what you mean?

The Viking Age is part of the medieval period.

Are you referring to the Christianization of Scandinavian peoples?

So, there are a few ways that you can approach a question like this. One of the classic approaches would be what archaeologists a culture history approach, where we look at the interactions between cultures to explain change over time. There are issues, of course, with this strategy, but it gives a relatively solid explanation for the early phase this particular process.

The ‘Viking Age’ as a discreet period in European history would have been a time of massive changes economically, politically, and socially within Scandinavia. While travel and trade had been an important part of Norse culture for centuries before the Viking raids began, this period being referred to as the Vendel Period, the expansion into the North Atlantic and Western Europe brought the Norse cultures into contact with the European monarchies and Christianity. The earliest contacts between Norse and Continental cultures were, of course, somewhat hostile (looking at you, Lindisfarne), it did not take long for trade to be established between Norse entrepreneurs and Continental ports. With trade comes the friendly spread of cultural ideas and technologies.

Also recall that the Viking Age was not only a period of raiding, but of settlement and integration. It was during this period that Normandy was established by those Norse that chose to stay on the Continent rather than return to Scandinavia, Iceland began to be settled, as well as several cities in the British Isles. Given the connections that these initial settlements had with the Scandinavian homeland, these settlements could effectively serve as conduits between Continental and Norse cultures. Later, of course, you have the rise of powerful monarchies in Scandinavia (think Harald Fairhair) and the unification of Norway, at which point you could argue that the transition from the Viking to the Medieval period was already well underway in Scandinavia.

Bogucki, Mateusz 2010 Viking age emporia around the Baltic Sea – a cul-de-sac of the European urbanization? In Making a Medieval Town: Patterns of Early Medieval Urbanization, edited by Andrew Buke and Mike McCarthy, pp. 151-165. University of Warsaw, Poland.

Fitzhugh, William W. and Elisabeth I. Ward (editors) 2000 Vikings: The North Atlantic Saga. The Smithsonian Institution, Washington D.C.

Helle, Knut (editor) 2003 The Cambridge History of Scandinavia Volume 1: Prehistory to 1520. Cambridge University Press, Cambridge.