Hi Zashi,
It sounds like you're having a great class. What you said about the mother wanting the little ones to Speak Speak Speak reminded me of one mother of a student I had who wanted the same thing. Her 5 year old was super shy so getting her to speak was quite a task. It's good to remember, like other people here have said, that that kids should be speaking at their pace, not their mother's.
I don't know if you want to pick a course book, but I enjoyed using the
Supertots book with this quiet youngster. It's full of bright coloured pictures that are simple enough to talk about and has reward stickers that she can add onto the pages herself. The workbook has lots of puzzles, games and colouring activities that the two of you can do together. The songs that match the book are also short and catchy with a lot of TPR, which means she can repeat the wordss if she wants to/ can, but if not she can do the actions and you know she's still absorbing the English.
One idea I got from the book was to use puppets. I used the photocopied characters from their book stuck to chopsticks, but any puppet or doll is great to bring to class. Bring one for you and one for her and then "let" the dolls have a conversation. It's a fun way to play in English.