Me is nominative here. So what it should be?
Bin ich dumm!
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The phrase "silly me" is colloquial English. The German equivalent, "Bin ich dumm!", is colloquial, too (dict.cc).
A more fitting translation would be "Ich Idiot" (or "Ich Schussel", like @Robert suggested), which would be a title you assign to yourself to indicate that you are indeed silly.