An emotionally healthy student does not bully other students. Because bullying is most likely to occur on school grounds, it follows that school districts would have the resources necessary to combat it. At present, they don't.
Administrators are limited, by training and authority, to punishing the bully. Allowed to continue is the toxicity in the bully's environment (created by family and friends) that led to him/her behaving as he/she did.
The administrator should have a psychologist to refer the bully to, who in turn would examine the bully's relationships in order to determine how he/she could best intervene. The bully needs psychological help. Chances are some people he has associations with need help also.
This help is expensive and may or may not be accessible. The school district should, at minimum, see that those caught bullying would meet, as a group, at least monthly to interact with and hear presentations from properly prepared psychologists. Cash-strapped school districts could have drives or ask for volunteers to help achieve this purpose.