How do beat King Ragnar

If you are referring to the Mount and Blade Games, King Ragnar has a renown of about 1200+. With that being said he can easily lead 100+ Troops with out having charisma or leadership skill. All in all that puts him in charge of about 240 troops that range from about a level 10+ "Nord Footman" to a level 28 "Nord Huscarl." Not only that but he is the King and therefore has vassals that will protect him and they generally have about 80 troops.
All Nord Vassals rely on "shock infantry" meaning they got some pretty tough soldiers for defending castles and such. However, the best way to beat them in a siege would be to have Swadian Knights or Vaegir Knights or their infantry counter part which i believe to be a Vaegir Guard or Swadian Elite Infantry unit. Basically the top teir for those factions. The Keghit Klan will not be able to do to well in a siege against the Nords but on the open feild they will dominate. Rhdoks and the New Faction (Yellow in Warband) you'd have to try and overwhelm which would be the hardest.

All in all to defeat the Nords you got to have a highly trained army, but that really goes for all kingdoms. However Swadians seem to be the best bet, followed by the Vageir.