This instance of Peirce's law uses the supplied classical double-negation elimination rule. First derive while assuming , using explosion inside a further temporary assumption.
1. (P → Q) → P          premise
2.   ¬P                 assumption [i]
3.     P                assumption [j]
4.     ⊥                ¬E, 2, 3
5.     Q                ⊥E, 4
6.   P → Q              →I, 3–5, discharging [j]
7.   P                  →E, 1, 6
8.   ⊥                  ¬E, 2, 7
9. ¬¬P                  ¬I, 2–8, discharging [i]
10. P                   ¬¬E, 9
This is a formal natural deduction proof from exactly the given premise; no unproved excluded-middle lemma is needed.

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