It makes no sense, unless you shift the entirety of the Americas to somewhere in the mid-Pacific.
1. Discovery? I don't know. China or Japan are the only options. Someone swept off course and returning (not dying/vanishing into North America like those castaways tended to), and the story surviving as a legend that makes its way into European lore/mapmaking that gets confirmed at some point. Even if the Asians don't know/tend to ignore what they've inadvertently discovered, the story could spread in Europe and be taken as fact by many like some medieval legends, but unlike some other stories, this one proven as true. Marco Polo or similar figure could be good to transmit the story. So that means Europe already "knows" of the New World by the late medieval period.
2. The Russians are in the best position to make use of it, if they could gain and keep Outer Manchuria before the 19th century. Since their harbours on the Okhotsk coast before then were pretty terrible. Other Europeans can't and have no reason to. China and Japan do, since the northern route is the quickest (albeit with very bad sea conditions) to the New World. But for Europeans?
We need somebody somebody pulls a Jermak before 1492 and conquers Sibiria. Maybe the Mongols push as far as the edge of Sibiria and are curious about the other side ? The rumors are spread over all Asia and eventually reach Europe the Priester Johannes(John) style.
There's nothing there but rather fierce reindeer herdsmen like the Chukchi. The Mongols don't need that many random steppe tribes paying tribute. And even then, there's nothing there in Alaska besides hunter-gatherers who are even poorer than the reindeer herdsmen.
Ah thank you Dathi, You have just come up with a new push to settle Vinland, the evacuation of Iceland during a massive volcanic episode.
If Laki didn't do it, then what would? They'd probably just go to the Faroes/Norway anyway.