The problem with the US buying Alaska is currently the Union's just straight up broke right now and is fixing a devasted country and will have to radically restructure its economy and trade what with it losing eleven (possibly twelve depending on what happened with Kentucky.) states, that's something that's going to take a long time, especially with a middling president like McClennan in charge. It could take a decade for America to recover but from what the Author told us, the new era is one of Hard Feelings, so the near future doesn't look too bright.
Russia's not gonna want to sell it for nothing and they certainly don't want the Brits to have it so they'll just sit on it and basically leave it to fend for itself, Britain for its own part isn't really gonna be interested in starting a war over Alaska so I can easily see Russia keeping Alaska well in to the 20th century depending on how WW1 turns out.
The issue is that Russia if it sits on the land it will lose it.
Unfortunately Russia cannot just hold onto Alaska, soon there will be thousands of Americans and Canadians streaming into Alaska for another gold rush. They will easily out populate the local Russian population (unless there a huge demographic shift I missed). Canadians and American settlers won't obey Russian law/Orthodox Religion. Though I do see a longer lasting Russian influence in the area, I cannot see it lasting into 1900. Also, 7 million still isn't that much for any of the American states, and it could be a good poltical win for an embattled President.
These are all big reasons why, sooner rather than later, Russia will feel compelled to part with Alaska to someone, preferably the US. Hanging on to Alaska has no serious long term gains from the Tsar's point of view, and it is indefensible by Russian arms, and all to easily seized by Britain in another potential war.
However, Russia will soon have another issue to consider on its Pacific slope that will dwarf the Alaska matter.