Ah, the vicissitudes of homeschooling young ones. Some days are simply a trial, aren't they?
It appears that you are dealing with a few different issues. First, young children can become overwhelmed when faced with too many unknown flash cards at once, especially if the session is too long, and then become discouraged. [Along with the teacher!]
A peek at the Little Stories for Little Folks Parents' Guide [the backbone of the program] suggests introducing no more than three or four flashcards at each session initially. Set aside no more than ten or fifteen minutes each day to work on letter names and sounds. Praise the children for correct responses, but don't dwell on the cards they don't know. [Please see Parents' Guide for more detailed, step-by-step instructions.]
Next, imagine showing a child who has never seen a cat, a single dimensional picture of a cat. Telling the child that this is a cat actually gives him almost no information about cats. However, once he sees, pets, and hears a cat meow, he understands what the photo represents. In a similar fashion, the Parents' Guide uses a teaching method that provides multi-dimensional learning that will make it so much easier for your children to learn not only the sounds, but also the letter names at the same time. This method of teaching both sounds and letter names at the same time saves work and frustration for both teacher and students. [For complete instructions, please see the Parents' Guide.]
By praising correct responses and following the steps and variety of presentations in the Parents' Guide, the children will master sounds and letter names much faster than by drilling on flash cards alone. As they master letters and sounds and move on to those that they don't yet recognize, they will lose that sense of 'Oh, no, more flash cards that I don't know,' and instead realize 'Hey, I can do these!'
May our good Jesus bless and guide your homeschooling days!