At utility scale you can have pumped hydro and other economic battery alternatives.
CATL's announcements of Sodium Ion batteries promise an order of magnitude cost reduction over lithium. Having enough storage to capture peak energy will be game changing; the only significant hurdle is to modernize the grid but there's a lot of resistance to that /rimshot
We could also choose to just build solar powered datacenters and have a follow-the-sun model for the data processing. That small delay would still meet most of the needs of users. But nope, BigCorp isn't smart enough to be that innovative and instead doubles down on the old method and demands more nuclear power.
Electricity is cheap compared to GPU capital costs and depreciation. Intermittent usage is only cost efficient with capital that cheaply buffers energy either generically like batteries or specifically like heating up your smelters. It is not unsolvable, but not as easy as you imagine to retool.
Why should data centers need more than the usual proportion of energy at night?
Well they'll end up with higher prices at night. I wonder if we'll see a shift to energy-intensive processes being run during the daytime.