It will solve itself. But I don't think the solution is nuclear. The solution will be storage.
Again, the issue is cost, and externalities. On the one hand the cost of nuclear electricity is low, but the externalities are extremely high. Time, public opinion, and long-term liability are all against it (and that's assuming it goes as planned.)
Sure, optimal storage has yet to be figured out. Batteries and pumped-water-schemes are working in some cases, but are not necessarily grid-scalable [1].
But work in this area has potential, and we've not reached any maxima here yet.
Technologies like compressed air, hydrogen and so on are sll in their infancy.
I should note that I am pro-nuclear. It's a lot cleaner than coal or gas. I'm just not sure it'll ever be economically attractive.
[1] on a household level, batteries that can store over-night amounts of power exist, send are easily available. This provides a cap on how expensive night electricity can be.