The city council of a city of the primary class shall have the power to grade partially or to an established grade, curb, recurb, gutter, construct sidewalks, or otherwise improve or repair any street or streets, alley or alleys, public grounds, public way or ways, or parts thereof, including sidewalk space, at public cost, or by levy of special assessments on the property specially benefited thereby, proportionate to the benefits. When the streets, public ways, or public grounds have been brought to an established grade, the city council shall have power to bring sidewalks and sidewalk space therein to a grade and to construct sidewalks and shall have power and authority to levy special assessments against the property specially benefited, not to exceed the cost of the improvement. Ordinary repairs, not including repaving or resurfacing or relaying existing pavement or making sidewalk repairs, shall be at public cost.