There was a lot wrong about the battle (on both sides, but to be fair the Night King and his disposable Other bodyguard doesn't seem to exactly have fine motor control over their wight minions, even the zoombie variant of the show, so there's only so many things they can get up to).
First and foremost, it was somewhat ill-advised not to bring supplies for the army. In fact, it might have been necessary to leave parts of the army in the south, waiting for Lannister supplies that will never arrive, as feeding the vast force depicted in the marching photos must have been a tall order.
Second, they had a giant light cavalry force (somewhat oddly, one that seems to have left all their bows at home) and a magical boy who can warg into flying animals. So a bit of scouting, perhaps?
Once you have done these two things, neither of which they bothered with, they've got a really strong defensible position in the castle and (from the information gained from their scouting parties, who probably could have done a bit of light harassment, possibly covered by a double dragon formation that keeps a lookout for the undead dragon) an enemy force that seems to mass up and assault from one direction. So time to start digging a lot of ditches, behind each other, positioning them to funnel the dead into each other in a big clump well suited for dragonfire raids, and making lots of dragonglass and incendiary arrows, as wights burn well. Considering how closely the Wights mass and how combustible they are, the trebuchets and 19th century misunderstanding of mangonels the Unsullied drag around could probably also be used to fire into the wight ranks, positioned on towers. Springalds should be plentiful in the walls and on the smaller towers, spesifically to bring down that enemy dragon. The ditches should be filled with whatever magical substance you seemingly have access to that can cause a simple palisade to burn for a looong time. And we're talking MANY ditches here. The walls should be reinforced and loaded up with stuff you can drop on the enemy, as should the gates, likely using that nice dragonglass surplus you have a lot of to make a spiky affair that is difficult to burst through. There should also be prepared positions inside the courtyard and on the central keep, enabling you to use a defense in depth effectively. The civies get put in the crypt and for chrissakes, you seal the tombs. Everybody knows Team Evil can raise the dead.
Since the Others seems to call up their blizzard only when necessary, it means you'd be unaware of it. So your Dothraki and Vale cavalry should probably plan for a flanking and harassament maneuver, relying on their speed to get out of the way of the dead wight-wall. If Winterfell has any sally ports, the cavalry might slip out there. If not, they'd have to stay on the move.
Your weirdo brother Bran's plan seem to be a good one. He can try to draw the Night King in and ensure he commits to his wasteful frontal assault.
Now the dead have to cross a myriad of burning ditches while being showered with burning pitch and the occasional dragon sweep (again, keeping a watch for the undead one, trying to draw it close enough to the walls to skewer it with a couple of springald spears, dragonglass-tipped or incendiary). If you can bring it down you will be able to really hurt the wight ranks with impunity. When they get close to the walls they get showers upon showers of arrows while crossing the burning ditches that you have a better way to light than trying to hit it with arrows. The sappers can probably fix that.
Ooops. Blizzard. Ok, luckily that also means that the enemy dragon can't see anything either and the trebuchets can just keep firing blindly, ranging from the last ditch you lit on fire with prepared ranges as it illuminates the horde. When the dead finally reach the walls you already have them manned, and can distrupt their circus human pyramids quite badly by dropping burning stuff and rocks on them. They can't really form those under bombardment from the walls. Most of your infantry should have a field day whacking zombies on the walls.
If the baddies break through, which they probably only will with a really depleted force, you have prepared positions to fall back on. It is really too bad about the cavalry, but at least you don't lose them in meaningless frontal assaults and if the storm breaks they can start hitting the flanks as the dead are funneled in.
This sounds like a plan. Your valyrian steel-armed dudes can hunt others, as can the dragonglass archers and dragons, if you can free them up by bringing down Dragon McZombieface.
Bring it on Mr. Thornhead.