Officers said Donna Brown was getting a hall ready for an AA meeting, when someone entered the building and attacked her.
"It's senseless," Galveston Police Chief Vernon Hale said. "So we need to find the culprit. We need to make sure we bring justice to the Brown family."
Police received a call around 4 p.m. Sunday for an unconscious woman in the 1800 block of 33rd Street.
Brown was transported to the hospital in critical condition. She died Thursday afternoon at UTMB.
Officers say Brown moved to Galveston a decade ago.
"I was astonished," neighbor Amanda Hornbeck said. "I mean, maybe in shock. I couldn't believe it. I mean, she's so careful."
Investigators are currently working to find any witnesses or surveillance images of the suspect.
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