For the quadrilateral shown, how many different whole numbers could be the length of the diagonal represented by the dashed line?

[asy]
draw((0,0)--(5,5)--(12,1)--(7,-8)--cycle,linewidth(0.7));
draw((0,0)--(12,1),dashed);
label("8",(2.5,2.5),NW);
label("10",(8.5,3),NE);
label("16",(9.5, -3.5),SE);
label("12",(3.5,-4),SW);
[/asy]
Label the vertices $A$, $B$, $C$, and $D$ as shown, and let $x = AC$.

[asy]
draw((0,0)--(5,5)--(12,1)--(7,-8)--cycle,linewidth(0.7));
draw((0,0)--(12,1),dashed);
label("8",(2.5,2.5),NW);
label("10",(8.5,3),NE);
label("16",(9.5, -3.5),SE);
label("12",(3.5,-4),SW);
label("$A$",(0,0),W);
label("$B$",(5,5),N);
label("$C$",(12,1),E);
label("$D$",(7,-8),S);
label("$x$", ((0,0) + (12,1))/2, N);
[/asy]

By the triangle inequality on triangle $ABC$, \begin{align*}
x + 8 &> 10, \\
x + 10 &> 8, \\
8 + 10 &> x,
\end{align*} which tell us that $x > 2$, $x > -2$, and $x < 18$.

By the triangle inequality on triangle $CDA$, \begin{align*}
x + 12 &> 16, \\
x + 16 &> 12, \\
12 + 16 &> x,
\end{align*} which tell us that $x > 4$, $x > -4$, and $x < 28$.

Therefore, the possible values of $x$ are $5, 6, \dots, 17$, for a total of $17 - 5 + 1 = \boxed{13}$.