You should try South Lanarkshire Council Cemeteries Office first of all, she is probably buried in Larkhall cemetery.
She must have died of some sort of fever or infectious disease if she was in Motherwell hospital (The County Hospital). It was a stones-throw from Airbles cemetery, so if she can't be found in any of the South Lanarkshire burial grounds your next bet would be Airbles, which is administered by North Lanarkshire Council.