In order to cover the entire cost of changing the grade or grading, as provided by sections 14-384 to 14-3,127, of any street, boulevard, highway, main thoroughfare, controlled-access facility, connecting link, major traffic street, alley, or part thereof, including intersections and damages awarded, a city is authorized to levy special assessments to the extent of the special benefits conferred by the improvement on the lots and parcels of land especially benefited by reason of the grading of any street or part thereof whether such property abuts on or is in the vicinity of the street or the part of the street so graded. All such special assessments shall be equalized, levied, and collected in the manner provided by law for the equalization, levying, and collection of special assessments. All grading shall be done to the full width of the street unless for good and sufficient reason the city finds that such grading shall be done to a different width.