its good you have a tank that size
are you feeding your fish pellets or flakes or what?
and if you said you dont have a heater, i highly recommend one. with your tank a mini 7-10 watt one will do. it highy improves the bettas healthy, and activity.
and it really isnt easy to tell how long cycling will take. if you do it right you will have (at the end) 0 ammonia 0 nitrites and less then 20ppm of nitrates. when i cycled my five gallon i think it took about 1 and half weeks.