While I can't comment on the intention behind this particular book, I would definitely not recommend it to children because it teaches centering prayer, which is being passed off as contemplative prayer but is actually based on Transcendental Meditation.
I was very disturbed after reading Fr. Jelenek's book which instructs children to select a "secret sacred word" (known as a mantra in the east) and to focus on this word while praying for six minutes every morning and evening. They are taught to use their sacred word to banish all thoughts from their minds, even though the Catechism specifically calls techniques such as this that blank the mind to be "erroneous notions of prayer." (No. 2726)
JD correctly claims to have found "no Catholic connection" within the book because Christian prayer is a dialogue with God, not a mind-blanking exercise. There is absolutely nothing in this book that encourages children to dialogue with God, to listen for His voice, to praise Him or thank Him or ask for His help. It only instructs them to "Silently say your secret word in your heart. Rest within. Sit and wait. God is there inside you, in the quiet. Rest within." (Do you really think children ages 3-10 understand what it means to "rest within"?)
Instead of teaching authentic Christian prayer, it focuses on blanking the mind, telling children to "Let your thoughts go. Forget them all. Let them float right out of your head."
Besides calling this kind of prayer "erroneous," the Catechism also says that prayer doesn't just come from ourselves, but from the Holy Spirit as well (No. 2726). What if the Holy Spirit has something to say, or some kind of inspiration or impression that He wishes to impress upon us? Do we have any hope of hearing Him when we're so actively working to keep our minds blank?
Anyone who understands the Catholic contemplative tradition, with its passive and active forms of contemplation, will know that mind-blanking techniques such as centering prayer are the fastest and most effective ways to prevent oneself from reaching higher stages of prayer - which only come at the invitation of God. How can this invitation come through when we're blanking our mind rather than learning how to be open and receptive to God in prayer, which is the most important prerequisite to authentic contemplation?
It's a true tragedy that children are now being taught this aberration, which can only result in shutting them off from the very God they are trying to reach.
Instead, we should be teaching them about the many different kinds of prayer such as vocal prayer, mental prayer, lectio divino (praying with Scripture) in order to open their hearts and minds to the many different ways we can speak to God and listen for His voice in the course of their daily lives - through the Word of God, sermons, the people we meet, interior inspirations, etc. Teaching them that contact with God happens only in silence (which this book suggests) cuts them off from having the kind of vital and authentic relationship with Him that they will need in order to live out their lives according to His will - which is the secret of true happiness.
Although my copy of the book does not have a Nihil Obstat, the publisher's website claims a Nihil Obstate and Imprimatur has been obtained from the Archdiocese of New York, which only complicates this picture even further. This is especially so because Fr. Jelenek bases his book on the work of Fr. Thomas Keating, the founder of Centering Prayer, and Keating's seminal book on the subject, Open Mind Open Heart, does not have an imprimatur (at least my copy doesn't). Fr. Jelenek is a member of Keating's Contemplate Outreach apostolate which teaches the practice of centering prayer throughout the world.
You may be interested in this blog about Fr. Kneemiller, who was once an enthusiast of the parent of centering prayer - Transcendental Meditation. He describes in chilling detail what kind of bondage he found himself in as a result of practicing this kind of prayer.
Our Learn to Discern series contains a very detailed and hard-hitting booklet about Centering Prayer that outlines its many flaws and clearly explains the difference between this and authentic Catholic contemplation.