If I have a 45 year old mentally ill brother, how can my mother get a POA over him?

I think what you might be referring to is a "mental health POA".  This is a written statement by a mentally person, known as the "principal", which grants powers to an "agent", to make decisions carrying out the principal's wishes on matters that exist/arise during an episode of mental illness. The mental health POA contains the principal's wishes about medications, hospitalization, and any other concerns that come up. It also typically contains language giving the power to deal with unexpected circumstances.
A mental health POA must meet certain requirements in order to be valid. While these requirements vary from state-to-state as a general rule it should be drafted and executed when the principal is mentally healthy and possesses the mental ability to understand the document and to intentionally and willingly sign it. The POA should expressly state that the agent's powers will not be activated until the principal is deemed incapacitated by a licensed psychiatrist or psychologist, etc. The POA will also contain a "revocation clause", which allows the principal to revoke the agent's power during a time when the principal is lucid, so that they can appoint a different agent or no longer wants a POA.
At this point, you should consult directly with a local attorney. They can more fully discuss all of this with you and provide guidance as to specific state law concerning POA's.