Its a very, very short list. Way too short to make a difference. In all of the North, we know for sure of two Valyrian weapons. Brienne, lurking outside Winterfell, has Oathkeeper one of the two blades forged from Ice, the Stark family sword, which Tywin Lannister had melted down. And Jon, of course, has Longclaw, originally the sword of Lord Commander Mormont. The dagger used by the unnamed assassin in the attempt on Bran Starks life in Season 1 was Valyrian, but who knows where that thing is now.
In the rest of Westeros, things dont look much better. Theres Heartsbane, wielded by Sams dad, Randyll Tarly in the Reach. Lady Forlorn is not only a country-pop trio, its also the name of the family sword of House Corbray of the Vale. Joffreys Widows Wail, the other sword forged from Ned Starks Ice, is now presumably in the possession of King Tommen, who wouldnt cut a piece of roasted venison without asking his mothers permission first. The tiny Iron Islands boast two Valyrian steel weapons: Nightfall is the family sword of the Harlaws, an ancient and powerful Ironborn house; and House Drumm hold Red Rain, which their ancestor Hilmar stole from some unnamed knight, thus earning Hilmar the sobriquet The Cunning. Finally, rumor has it that tiny and ancient House Celtigar have, among the many treasures overfilling their castle on Claw Isle, an ax of Valyrian steel.
Additionally, there are a half-dozen notable Valyrian weapons whose whereabouts are unknown. Most famous of these are the original Lannister family sword Brightroar, lost when King Tommen II disappeared in Valyria back in the day; Blackfyre, the sword of Aegon the Conqueror, which is probably in Essos somewhere; and Dark Sister, the sword of Aegon the Conquerors more martial sister-wife Visenya, which was last known to be held by the Targaryen great bastard Brynden Rivers. Brynden was sent to Castle Black back in 233 AC, and he eventually rose to Lord Commander. If Lord Brynden was allowed to take the sword when he took the black (which seems highly unlikely), theres a chance he may still have it with him under the great weirwood far beyond the Wall.
As for making more Valyrian steel, dragonfire is often mentioned as being part of the process used to make the weapons, along with a careful folding of the metal and treating it with some kind of magic. So, while the reappearance of dragons theoretically makes forging new weapons possible, because of access to their fire and the reawakening of magic tied to the rebirth of the beasts, its the spellcasting part of the equation that remains a problem. Notably, the Targaryens in the roughly 150 years of their nearly 300-year reign in Westeros during which they possessed multiple dragons never forged a new Valyrian weapon. Considering the rarity of the weapons, it seems certain that they wouldve forged more steel if they knew how. Obviously, the spells used to create Valyrian steel were not well known, even among the Valyrian elite.