jmc

We don't know $g(x)$, so we don't have an expression we can simply stick $8$ in to get an answer. We do, however, know that $g(f(x))=2x+3$. So, if we can figure out what to put into $f(x)$ such that $8$ is output, we can use our expression for $g(f(x))$ to find $g(8)$.

If $f(x)=8$, then we have $x^2-7x+18=8$, so $x^2-7x+10=0$, so $(x-2)(x-5)=0$ which means $x=2$ or $x=5$. Since $x$ could be $2$ or $5$, we could have $g(8)=g(f(2))$ or $g(8)=g(f(5))$. Using the given expression for $g(f(x))$, the two possible values of $g(8)$ are $g(f(2))=2\cdot 2+3=7$ and $g(f(5))=2\cdot 5+3=13$. The sum of these is $7+13=\boxed{20}$.