Fishless cycling will take a huge ammount of ammonia in 200 gallons of water. If you can see if your lfs will sell you some biological filter media from one of their filters, if they will then all you have to do is place it inside one of your filters and you have a instant set up that you can add your existing fish to right away. Usually you can just do this from tank to tank but the size difference between 20 and 200 gallons may be too great for the bacteria from your old filter to have any effect.