The phrase "silly me" is colloquial English. The German equivalent, "Bin ich dumm!", is colloquial, too (dict.cc).
However, "Bin ich dumm" would be improper grammar, since sentences beginning with a verb should generally be questions. Sentences beginning with a verb can be intonated so that they sound like requests (Aufforderungssatz), for example, "Wirst du das wohl lassen?/!".
Therefore, aA more fitting translation would be "Ich idiot"Idiot" (or "Ich Schussel", like @Robert suggested), which would be a title you assign to yourself to indicate that you are indeed silly.