Shortest path that passes through specific node(s)
Compute the shortest path between all pairs of the twelve special nodes (nodes 1, 100, and all ten of your particular nodes), and use these as edge lengths in a new graph consisting only of these twelve nodes. Now, you have a graph containing twelve nodes, and you want to find the shortest path from 1 to 100 that uses at least five other nodes. Now, you can use your algorithm (which I think is just dynamic programming) on this graph, which is much smaller. The solution on the new graph will give a solution for the original graph, which you can find by replacing the edges in the solution in the new graph with the shortest paths you computed in the first step.  

For your example, we will have 4 nodes in the new graph. The edge lengths in the new graph are: 0-1:5, 0-4:6, 0-5:11, 1-4:5, 1-5:10, 4-5:5. These correspond to the shortest paths between nodes $i$ and $j$ in the original graph. The shortest path including one node from the list is 0-4-5, which has length 11. Now, replacing each edge in this path with the path computed in the first step gives the path 0-3-4-5 in the original graph.