What is the ruling concerning the 'Umrah of my young daughter who did not perform the rituals due to fear of illness and exhaustion, knowing that we stipulated desisting (تحلل) should an impediment arise?

If you performed 'Umrah on behalf of your daughter and then did not complete it, there is no sin upon you, because the ihram of a child is not binding due to their not being accountable. This is the view of the Hanafis and has been chosen by some scholars. As for stipulating a condition, it is beneficial if there is an impediment or an obstruction, such as illness, but mere fear of illness or exhaustion is not sufficient.