Wind and solar cannot provide all our energy. Nuclear does not replace wind and solar, it complements it. The sun doesn't always shine, and the wind doesn't always blow.
Yes it costs more than burning gas or coal, but that doesn't take into account the environmental cost or the cost on the health of living things.
And yeah we do know how to store nuclear waste for *hundreds of years. We do so already. I don't know where you're getting the "hundreds of thousands of years" from. Even old reactors althat don't recycle waste didn't have half-lives anywhere near that long.