The story describes Jamie as a righteous and responsible man, even appointed so by his brother, Ted, during their chat with Carmody. Jamie's nature drove him to tell the truth about the incident over lying for the reputation of his brother; he knew it was the best decision for him, Ted, and the victim.
Having given light to the truth:
The victim would rest assured on account of the culprit's arrest - even though he owes great responsability for the incident as well, as he was drunk.
Jamie would not have to deal with deception staining his righteous principals for the rest of his life,
Given Ted's inability to make reasonable decisions in times of great stress, it's better for him to let the truth be known swiftly, rather than allowing the guilt to build up over time only to end up snapping grievously with the final confession, and therefore, having to face a more serious penance in the future.