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The Blueprint LG book has every LG game rated for difficulty. So I try to do at least one super hard game most days. I also drill by type as Blueprint has that marked for every game too.

My question is how soon I should check an online or written explanation for missed inferences if a particular game is giving me difficulty? Is there benefit to staying in that hard place and working to try and see if the missed inferences will come? Or better to just see as many different LG's as possible and keep turning them over?

I'd say that if you can't figure it out timed, try it untimed. If you can't figure it out untimed, you need to learn a way of doing it. Then, after the explanation, work through it (even though you'll already know what answers are right, think of it like a physical exercise, you're just practicing your form) then, a couple days later (hopefully you've done a bunch of LGs in the interim and you'll have forgotten which answers were right, otherwise wait a week) do it timed again and see if you've learned your lesson.

Just my two cents, when I personally have the discipline to do this is when I see most improvement...