After the contractor has fulfilled his contract to the satisfaction of the street superintendent, or the engineer if such power has been delegated to him, the superintendent or engineer, as the case may be, shall forthwith make an assessment to cover the sum due for the work performed and specified in the contract, including all incidental expenses, in conformity with the provisions of this part according to the character of the work done, and in the event an acquisition is included in the proceeding, including the amount then estimated by the street superintendent or engineer as the case may be, as the cost of the acquisition.