After Lord Jon Arryn's untimely demise, King Robert of the House Baratheon, the First of His Name, King of the Andals and the First Men, Lord of the Seven Kingdoms and Protector of the Realm, decides to make Eddard Stark, Lord of Winterfell, the new Hand of the King. He furthermore chooses to not just send a raven to summon Ned to court but travel to Winterfell with half the court, including the Queen and half her family as well as their children.
We also know that the distance between King's Landing and Winterfell is quite large, albeit the numbers are all but certain (Yoren of the Night's Watch claims it's 1,000 leagues). So for all we know, they will therefore spend quite some time on the Kingsroad going there and back again.
With the King gone and no new Hand appointed, who will rule the realm and speak justice in the name of the King? Is any of the members of the small council announced to chair the small council and to serve as the King's proxy during his absence (powers that would otherwise rest with the Hand of the King)? Is there a Hand pro tempore? There is, after all, a precedence for such a thing with Tyrion Lannister serving as acting Hand in his father's stead under King Joffrey.
It is clear that King Robert was more interested in hunting and whoring than in ruling the kingdom and therefore let the small council run the realm in his stead. Still in those days there was Jon Arryn presiding the council, a man that King Robert would trust.