How can I atone for the the sin of taking my mother’s ring and losing it without her knowledge when I was twelve years old?

You must be frank with your mother and ask for her forgiveness, and compensate her with a ring similar to hers. Whoever takes another's property without permission is a usurper and is liable for compensation, and young age does not negate this liability. If it is highly probable that frankness will lead to a rift, then it is sufficient to give her a ring, even if ostensibly as a gift.