If the application is not filed in the office of the board within 30 days after completion of the giving of notice by publication, or if the applicants fail to present to the board and obtain its approval of the plans and specifications and to commence the work within 60 days after filing the application or any further time the board may allow, or fail to complete the work with reasonable diligence, the board may proceed with the construction, repair or completion of the work, and pay the cost by assessment upon the lands within the drainage district directly or indirectly benefited by the levee according to the benefits, as provided for in this part.