The particle indicates purpose. (It's not .) []{} here is a noun that means []{}[]{}, "shaking off one's sleepiness". I think is more like "First (of all)," and modifies , not .
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_Lit._ "First of all, for (the purpose of) shaking off my sleepiness, I'll just take a shower and..."
→ "First of all I'll just take a shower to shake off my sleepiness and..."