。
She didn't mind washing the dishes.
Literal
She [topic-は] dishes [object-を] washing [nominalizer-の] [object-を] didn't-show-reluctance.
嫌がらなかった is the negative past of 嫌がる (いやがる, 'to show reluctance, to be unwilling'). Like other ~がる forms, it describes observable behavior rather than internal feeling. 食器 (しょっき, 'dishes, tableware') is the general term for plates, bowls, and utensils.