Here is my problem,

I just bought a refrigerator that has an ice maker and water dispenser which by the way is a Haier PRCS25. I then purchased an ice maker kit from the hardware which came with a self piercing saddle valve. From here I connected the valve to a 3/4" copper cold line pipe running horizontally just bellow the floor of the refrigerator. Once I pierced the hole in the pipe and backed the pin out I was able to make ice but water was just dripping out of the dispenser (not at all in a steady stream as I would expect). After this I drilled a 1/4" hole where the pin pierced the pipe and I received the same results. Also I have city water so the PSI is around 50.

Would it help to reattach another valve to the vertical line running up the wall behind the refrigerator? Any professional suggestions would be greatly appreciated to help me fix this problem without calling a plumber.