Note that: \begin{align*} 1\cdot 1 = 1&\equiv 1\pmod{13} \\ 2\cdot 7 = 14 &\equiv 1\pmod{13} \\ 3\cdot 9 = 27 &\equiv 1\pmod{13} \\ 4\cdot 10 = 40 &\equiv 1\pmod{13} \\ 5\cdot 8 = 40 &\equiv 1\pmod{13} \\ 6\cdot 11 = 66 &\equiv 1\pmod{13} \end{align*} Thus the modulo $ 13 $ inverses of $ 1,2,3,4,5,6 $ are $ 1,7,9,10,8,11, $ respectively. The only residue from $ 7 $ through $ 12 $ that is not the inverse of a residue from $ 1 $ through $ 6 $ is $ \boxed{12} $ (which is its own inverse). (Note that $ m-1 $ is always its own inverse modulo $ m $, so we know without doing most of the work above that Dayna could not have erased $ 12 $. The rest of the work only goes to confirm that $ 12 $ is the $ \textbf{only} $ residue she did not erase.)