How often were you changing and cleaning the tank with the old fish? 2 gallon tank for a 6 inch goldfish is quite small i think. You would need something like a 10 gal tank for that fish to give it room to swim and for the ammonia levels to not spike to big levels. Someone will correct me if I'm wrong.
I think this will have mixed reviews as to whether or not you have to cycle your tank but if you want to be safe then cycling is always a safe bet. But it means getting a chemistry set to watch for ammonia, nitrite and nitrate spikes, changing a quarter of the tank water once a week or even daily to make sure the levels stay low and so on.
Other choice you could do (and my younger sister does this) is you change the water weekly and clean everything weekly including the gravel. She rinses it off in hot water and scrubs the decorative plants and chest she has. This makes sure that her Betta is never in a bacteria filled world and so there is nothing to harm him.
As how the length, Cycling can take anything as short as 3 weeks to something long like 2 months. My 20gal tank took 1 and a half months. I never knew of the fishless cycle at the time and got hardy fish right away. The benefit to this is that the tank cycles while you already have some fish it but they have to be hardy to withstand the ammonia spikes. Some fish are really sensitive and can die quite quickly if they don't enter an established tank. If you want to stay to tropical fish, Some people will disagree with me, but a couple of male Guppy's would be perfect for your tank. They don't get any bigger than about 1-1 1/2 inches and they are really colourful and enjoy swimming too. They are quite hardy for a new tank tho some people say otherwise.
If the fish in cycle is no good for you check this link out by iamntbatman for other options as to cycling your tank. http://www.tropicalfishkeeping.com/f...m-cycle-38617/
He helped me out when I first started here and hopefully he doesnt mind that I share his link with other people. Goodluck!
PS. if you plan to stay with goldfish then don't put tropical fish into your tank. They produce different kinds of waste and they create different kinds of bacteria in the tank.