The answer is $6$. Clearly, all further lengths must be at least $8$ and at most $16$ to form a non-degenerate triangle with $5$ and $12$. In particular, $5$ is the smallest length. Now, if $n$ is the next smallest length, then the largest length can be at most $n+4$ to form a triangle with $n$ and $5$. Hence, there can be at most $5$ lengths that are distinct from $5$, and hence at most $6$ in total. Finally, clearly $6$ is possible e.g. by choosing $5,8,9,10,11,12$ or $5,12,13,14,15,16$ (or indeed $5,n,n+1,n+2,n+3,n+4$ for any $8 \le n \le 12$).

Answer --> 6