Function $g$ is defined by $g(x)=-k^x+m$, where $k$ and $m$ are constants. In the $xy$-plane, the graph of $y=g(x)+6$ has a $y$-intercept at $(0,\frac{29}{5})$. The product of $k$ and $m$ is $\frac{36}{5}$. What is the value of $k$?