Why did Sauron attack in the North as well?

[Earendilyon]

Well, Sauron was indeed strong enough to attack on all fronts at the same time, but for some things he had not counted upon, nor could have. First, his most powerfull minion, the WitchKing was slain, which he had not counted on. Even without the WK, his troops would have been strong enough to conquer Minas Tirith, but for the extra forces which Aragorn brought in. Those forces were at first occupied in the South, but were unexpectedly helped by the Undead Aragorn brought into the game. Again, something highly unforeseen.
Even with these losses, Sauron was strong enough to easily wipe out the forces gathered before the Black Gate. Remember, that the forces of the Free Peoples were almost overcome (even with the help of the Eagles), when the thing happened Sauron never had foreseen, not in his worst nightmares: the Ring was thrown/fell in the Cracks of Doom!
As you said yourself, the forces in the North were barely overcome by the peoples of Dale and Erebor. They were only overcome after the Ring was destroyed, and even then with much difficulty.
While all this was going on, forces from Dol Guldur were attacking Lórien. I don't think Sauron expected that those forces could overcome the Elves of Lórien, but they could keep them (and Galadriel!) occupied. He thought to overcome them in a later stage of the War. And then: to Rivendell!