Taye is expected to be charged with manslaughter, driving with a revoked license, and leaving the scene of a crash.
Smith died at Christiana Hospital Monday morning. She was taken to Christiana in critical condition following the Saturday night crash and her condition worsened Sunday night.
The Delaware City EMS worker was treating a motorcycle driver on Route 13 near School Lane Saturday about 10 p.m. Thirty-year-old Edward Reiss of New Castle, had crashed his Harley.
Smith was caring for Reiss when a 2004 BMW 760 sideswiped a police car, narrowly missing the officer, and ran over Smith and Reiss. Reiss was hospitalized with serious injuries.
Police say a car following the BMW stopped and let the BMW driver get inside, before speeding off.
The Associated Press contributed to this report.