Step 1: Find where the inside changes sign.
For $f(x)=\left|\dfrac{x}{3}-3^{-x}\right|$, set $\dfrac{x}{3}=3^{-x}$. By trying $x=1$: $\dfrac13=\dfrac13$. So the crossing point is $x=1$.
Step 2: Decide the sign on each side.
For $0\le x\le1$, $\dfrac{x}{3}\le 3^{-x}$, so $f(x)=3^{-x}-\dfrac{x}{3}$. For $1\le x\le5$, $f(x)=\dfrac{x}{3}-3^{-x}$.
Step 3: Split the integral.
\[ \int_0^1\!\Big(3^{-x}-\tfrac{x}{3}\Big)dx+\int_1^5\!\Big(\tfrac{x}{3}-3^{-x}\Big)dx. \]
Step 4: Separate the rational and log parts.
The $\dfrac{x}{3}$ pieces give the clean number $K=\dfrac{23}{6}$. The $3^{-x}$ pieces give a term with $\log 3$.
Step 5: Match the log part.
The $3^{-x}$ pieces combine to $\dfrac{82}{3^5\log 3}=\dfrac{3^4+1}{3^5\log 3}$, so $n=4$ (since $3^4=81$ and $81+1=82$).
Step 6: Compute $nK$.
\[ nK=4\times\frac{23}{6}=\frac{92}{6}=\frac{46}{3}. \]
\[ \boxed{\dfrac{46}{3}} \]