IIRC (and I may not) the "signs" in The Miracle Worker are limited to the manual alphabet. In the actual language ASL, the manual alphabet is used for about 15% of the total sign vocabulary -- mostly used for the spelling of proper names and titles, but occasionally, letter handshapes are incorporated into a sign (such as using the "U" handshape for "uncle" and the "a" handshape for "aunt").
So using the alphabet for entire conversations would kind of be like chatting in Morse Code ;-)
(It's useful for the deaf-blind, though, because each letter can fit into the palm of the recipient's hand).