Are you stuck on reading the results of the ssh call, or sending them to the client? You didn't say. Also, why does the server script call the run_script using ssh, if they are on the same machine? Surely you could skip that part and just call the run_script directly? (Using backticks or system, backticks will get you anything the script prints to STDOUT, eg: my $result = `run_script`;.
To return it to the client, you just need to print that information to the socket.