With one ear on the sound monitor linked to his daughter’s room, Joe Trimarchi sat at his dining room table in the Rosebank section of Staten Island. He was stuffing dozens of envelopes, hoping that one of them would land on the desk of a politician who might care. Mr. Trimarchi, 55, is a cashier in the billing department of Staten Island University Hospital, but what animates him is advocating for his daughter Alexia.