Brent does not admit that he tried to kill himself first of all because he is too deeply ashamed. He is basically a good person who made a stupid, tragic mistake with horrible consequences. When he becomes aware that his foolishness in drinking and driving has resulted in the death of an innocent human being, he can't face that fact, and somewhere in his mind, to add the stigma of a suicide attempt to all that has already happened is more than he can handle. Brent finds "the fact that he'd tried to kill himself impossible to share with another soul". It is too deeply personal, a perceived weakness within himself that he cannot reveal.
Brent also does not admit that the accident occurred largely because of a suicide attempt because he barely believes it himself. He can "scarcely believe he'd actually tried it and wondered how he could have given no thought to the other cars he would hit". Brent is an intelligent person, and if he had not allowed his judgement to have become so impaired, he would have never done anything to hurt an innocent person. Even now, after the fact, he is having a hard time believing that what has happened is real.
Brent is very sorry for having been the cause of the accident that killed Leah, and lives with a huge burden of guilt. Although he is terrified at the thought of going to prison and is relieved to find out that he will not, he realizes that he wants to be punished, to find a way to make restitution. It is possible that, subconsciously, Brent is aware that revealing that he had tried to kill himself would put more focus on his disturbed mental state and relieve him of a little of the responsibility for what has happened. Brent does not want that; he only wants to make amends (Chapter 3 - "The Afterlife").