Me
is nominative here. So what it should be?
Bin ich dumm!
or
Dumm ich!
@Lennart: Just because a sentence starts with a verb doesn't mean it should be a question. "Spiel nicht mit deinem Essen!" "Wollte er es, hätte er es gemacht."
Ich würde "silly me" mit "Ich Schussel" oder "Ach, natürlich..." ("... muss ich es so machen!") übersetzen.
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.