Los Angeles will build a new jail to house mentally ill offenders. The plan, approved by the county’s board of supervisors on Tuesday, will serve thousands of people in an environment focused on mental-health treatment. “The old way of doing things simply isn’t working and isn’t just,” the county’s district attorney said. The new facility will have 3,885 beds and more open spaces for guards to monitor inmates. Analysts estimate about 20 percent of L.A.’s 17,000 offenders are mentally ill.