I totally sympathize with you. Hong Kong parents must be the most driven people in the world when it comes to having their kids excel in English acquisition, and I've been on the wrong end of several family feuds over it. Teaching a three-year-old anything at all should be fun. Small children learn through play. Give them toys and the freedom to ask questions, and they will invent and learn till they fall asleep. The trick is to channel their energy and interests, as you have discovered. Here are a few things that I have learned over the years.
You say that toys and books worked for a while, but now your student has lost interest. That happens usually when the child has become bored. You have to keep things constantly moving and changing in order to keep short attention spans on target. Obviously, you can't constantly provide new books and toys, so you need to use what is available in different ways.
Let's say you want to teach/reinforce basic counting skills. Books may be a starting point, with plenty of colorful things to count, but books also are predictable and easily memorized. Therefore, children will quickly become bored with them. (You might actually count the books, but that won't work more than once.) Once the child understands 1, 2, 3, etc. move on to the toys. As duplos or tea sets are taken out of the box, or put away, count with the child. Count fingers and toes, providing plenty of physical contact and laughter therapy along the way. Skip or hop around the room, counting as you go. Sing songs that require counting, such as "One Little, Two Little, Three Little Indians".
Children are full of questions. But when they ask a question, they want to understand the answer right now. Do you speak your student's mother tongue? If so, then try to answer questions so that she understands the answer. Then have her practice asking the same question in English, and give her the English answer.
Try to enter as much as possible into her world. Whatever activity she is doing, do it with her, and use it as a catalyst to teach/reinforce English. Ask her to show you how to do something. Children love to learn, but they also love to show off. Once you know that she has mastered one thing, you might make deliberate mistakes and let her correct you.
I have set up a play store and allowed students to "buy" things using monopoly money. This can teach a host of skills, such requesting things, counting, colors, following directions, etc. You can use the toys she has for your store, but children alwas love to see and handle new things, so be prepared to bring things to class with you. When it is time for you to pack up, you can also teach her the concept of "mine" and "yours".
How long do you spend with your student each lesson? Try to limit the time to managable segments for her. And always leave her wishing that you didn't have to leave so soon.
Hope this helps.
Brighten the corner where you are.