There was essentially nothing left to rule over. The Witch King came to the North around TA1349 with the sole intention of destroying Arnor, or rather the three kingdoms that formed when Arnor split. With over 600 years of war, he achieved exactly that. Cardolan was destroyed and depopulated, and Rhudaur's Dunedain population was gone by TA1409. Arthedain had always been the most powerful of the three kingdoms but it was small compared to Gondor; see for example the reaction to Earnur's fleet:
... munition and provision for a war of great kings. Or so it seemed
to the people of the North, though this was but a small sending-force
of the whole might of Gondor.
Thus when the Witch King "captured Fornost, and drove most of the remaining Dunedain over the Lune" ... "the North-kingdom ended, for the Dunedain were now few, and all the peoples of Eriador diminished."
All quotes and information taken from LotR appendix A I parts (iii) & (iv).
As others have mentioned, there is evidence of small settlements scattered around Eriador (e.g. chapter 2 of the Hobbit), but there were no major cities or fiefdoms left; the North-kingdom was gone.
Tokien says very little about the later repopulation of Arnor, but there is a conversation between Gandalf and Butterbur in Homeward Bound (LotR book 6 chapter 7) that sheds some light on the matter. Gandalf remarks that
Indeed the waste in time will be waste no longer, and there will be
people and fields where once there was wilderness.
And many folk used to dwell away north, a hundred miles or more from
here, at the far end of the Greenway: on the North Downs or by Lake
'Up away by Deadmen's Dike?' said Butterbur, looking even more
dubious. 'That's haunted land, they say. None but a robber would go
'The Rangers go there,' said Gandalf. 'Deadmen's Dike' you say. So it
has been called for long years; but its right name, Barliman, is
Fornost Erain, Norbury of the Kings. And the King will come there
again one day; and then you'll have some fair folk riding through.'
The implication is clear: the land is currently empty, and
Gandalf expects settlers to move north from Gondor. After all, it
seems likely that the population of Gondor would increase during the
fourth age, in the absence of a powerful enemy waging relentless war