From the reading of The Brothers Karamazov, please give me an idea of how to respond to Dostoevsky's question: "If you could assure that your torture of one little child would result in human happiness and peace and contentment for everyone else in the world for all time (that’s what destiny means after all), would you agree to do it?"

To begin, I would note that the thought experiment involving the tortured child actually appears as the culmination of a much longer argument on the part of Ivan Karamazov, in which he details his grievances with Christian ideals and the Christian God. In a way, I'd suggest that the question you pose: "would you agree to this contract?" actually misses some of the deeper questions which Ivan is asking, because underlining Ivan's entire argument here is the observation that, ultimately, the torture's already happening. Human society and civilization is rife with suffering, exploitation, and cruelty on unimaginable scales. His question is: can any kind of providence ever justify that suffering?
This is an attack on the fundamental claims of Christianity: that the universe is fallen and will be redeemed. Ivan answers that the cruelties and suffering present in the world can never be redeemed. If human suffering is the price for paradise, if even a single innocent is required to suffer in order to bring it about, then the price is too high.
Keep in mind, then, that Ivan's not necessarily asking whether you would personally sacrifice an innocent child to torture and unimaginable suffering for the well being of everyone else on Earth. To frame it in those terms underestimates the degree to which this misery's already present in the world. Ivan's question is: can you accept a world where this kind of calculus serves as the foundation for your own happiness and well being, as well as that of everyone else's? Can any amount (even infinite amounts) of happiness counterbalance the suffering which purchased it?