We know that $g(f(x))=2x+3$, so we want to find $g(8)$.
To find $g(8)$, we need to find the value of $x$ such that $f(x)=8$.
Setting $f(x)=8$, we have $x^2-7x+18=8$.
Rearranging, we get $x^2-7x+10=0$.
This quadratic factors as $(x-5)(x-2)=0$.
So the possible values of $x$ are 5 and 2.
Now we can find the corresponding values of $g(8)$.
For $x=5$, we have $g(f(5))=g(8)=2(5)+3=13$.
For $x=2$, we have $g(f(2))=g(8)=2(2)+3=7$.
Adding these values, we get $13+7=\boxed{20}$.
The answer is: 20
Below is a math problem. Provide a detailed, step-by-step solution.

### Problem:
Let $f(x)=x^2-7x+18$ and let $g(f(x))=2x+3$. What is the sum of all possible values of $g(8)$?

### Answer:
