Even with the Great Game as per OTL, if the USA declines to buy Alaska, the Russians may be inclined to sell Alaska to Britain anyway. The Great Game has been interpreted in various ways (both at the time and in retrospect), and there were certainly times when a sale could have occurred, namely times when Britain and Russia actually colluded ("You take this, I take that, and we quitly accept each others' gains...").
Even if not, the Great Game - such as it was - is considered to have ended in 1895. The Klondike gold rush started a year later. This evidently increased the value of Alaska, if only because of getting sea access to the Yukon. It is conceivable for a sale of Alaska to occur in 1895, yet also note that the Klodike gold rush was started by the discovery of gold by... an American. Who would be less likely to be there if the USA never bought Alaska. So you can easily see a scenario wherein Alaska is sold to Britain at some point not long after 1895, and Canadian prospectors discover gold in the Yukon. Which at once makes Alaska more useful to them!
This is also where the major impact on history starts. Alaska in Canadian hands during an alt-gold rush is economically better for Canada. From there on, the further discoveries of natural resources in Alaska will continue to benefit Canada, which is nice for Canada. This is hardly world-shattering AH here, but it produces a TL that's economically good to the Canadians, at least.