Immediately after its completion the diagram shall be delivered to the superintendent of streets, who shall immediately, after the contractor has fulfilled his contract to the satisfaction of the superintendent of streets or the legislative body, on appeal, proceed to estimate upon the lots or parcels of land within the assessment district, as shown by the diagram, the benefits arising from such work, and to be received by each such lot or parcel of land. He shall thereupon assess upon and against the lands in the assessment district the total amount of the costs and expenses of such work, and in so doing shall assess the total sum upon the several lots or parcels of land in the assessment district, benefited thereby, in proportion to the estimated benefits to be received by each of the said several lots or parcels of land.