jmc

Rewriting $\frac{4321}{5^7\cdot 2^8}$ as a decimal with a denominator of $5^8\cdot 2^8=10^8$, we have $$\frac{4321}{5^7\cdot 2^8}\cdot \frac{5^1}{5^1}=\frac{4321\cdot 5}{10^8}=\frac{21605}{10^8}=\mathrm{0.00021605.}$$So, the sum of the digits of the decimal representation is $2+1+6+0+5=\boxed{14}$.