16. (12 points) In the Cartesian coordinate system $x O y$, the graph of the linear function $y=k x+b+2(k \neq 0)$ intersects the positive $x$-axis and $y$-axis at points $A$ and $B$, respectively, such that
$$
\mathrm{S}_{\triangle O A B}=|O A|+|O B|+3 \text{. }
$$
(1) Express $k$ in terms of $b$;
(2) Find the minimum value of the area of $\triangle O A B$.
16. (1) Let $x=0$, we get $y=b+2(b>-2)$; let $y=0$, we get $x=-\frac{b+2}{k}>0(k>0)$.
(2) From (1), we have
$$
\begin{array}{l}
S_{\triangle O A B}=\frac{(b+2)(b+5)}{b}=\frac{b^{2}+7 b+10}{b} \\
=b+\frac{10}{b}+7=\left(\sqrt{b}-\sqrt{\frac{10}{b}}\right)^{2}+7+2 \sqrt{10} .
\end{array}
$$

When $\sqrt{b}-\sqrt{\frac{10}{b}}=0$, i.e., $b=\sqrt{10}$, $S_{\triangle O A B}$ has the minimum value $7+2 \sqrt{10}$.