jmc

Label the vertices $A$, $B$, $C$, and $D$ as shown, and let $x=AC$.



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

By the triangle inequality on triangle $CDA$, $$\begin{array}{rl}x+12& >16,\\ x+16& >12,\\ 12+16& >x,\end{array}$$ 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 $17-5+1=\boxed{13}$.