I just replayed this part.
If you have done everything before this by checking the computer room above the complex, you should just be able to click the rope to the mine.
Then he will say 'where will I tie the other end to'. This time you click the rope on the rope attached to the blimp.
After this - an auto scene is seen - the complex is destroyed.
If you do not see this - it means that you have not completely done everything in the computer room.