jmc

We see that $8^4=4096$ is the largest power of 8 less than 8888, and that it can go into the given number 2 times, leaving us with $8888-(2)(4096)=696$. The next largest power of 8, $8^3=512$, can only go into 696 once, giving us the remainder $696-512=184$. Since the largest multiple of $8^2=64$ less than 184 is $2\cdot 64=128$, we are then left with $184-128=56$. $8^1=8$ goes into 56 exactly 7 times, leaving us with a remainder of $56-56=0$ and subsequently a coefficient of 0 for the $8^0$ term. So $8888_{10}=2\cdot 8^4+1\cdot 8^3+2\cdot 8^2+7\cdot 8^1+0\cdot 8^0=21270_8$, and the sum of the digits is $2+1+2+7+0=\boxed{12}$.