Whenever the board of directors of any district organized under sections 245.285 to 245.545 shall determine that the levees constructed by the district are insufficient from any cause to afford complete protection from overflows to the lands in the district, and whenever any such board determines that the construction of an additional levee or levees or the enlarging and strengthening of the existing levee or levees within the district would also afford additional protection to lands situated in another levee district or districts organized under sections 245.285 to 245.545, the board of directors may proceed to cause surveys, maps, and profiles of the proposed work of improvement and estimates of the cost of the work to be made by some experienced and competent engineer who shall make a written report, showing the amount, character, and kind of work to be done, the exact location and probable cost thereof, and return the same with all plans and specifications to the board of directors. Upon receipt of the report of the engineer and approval thereof by the board of directors, the board may direct that copies of the report be forwarded to the board of directors of each such other levee district, with the request that each such other board meet in joint session with the board of directors of the district in which the improvement is to be constructed for the purpose of determining the proportionate share of the cost of the improvement which should be borne by each district involved. When all such boards, in joint session, shall have agreed upon the proportion of the cost to be borne by each district involved, the board of each district shall thereupon proceed in the same manner as is now provided in section 245.480 to call a landowners’ meeting and to levy an additional tax and an additional percent of the assessed benefits expended for such purpose. Any proposal to levy an additional tax as provided in this section shall be approved by a majority in each district involved to become effective. The proceeds of any additional tax or the proceeds of any bonds which may be issued by any district for the purpose authorized in this section shall be transmitted to the treasurer of the levee district within which the contemplated improvements are to be constructed, and shall be placed in a separate fund and shall be used for no other purpose than to construct such improvements. Any sum remaining in the fund after the completion of the improvements shall be returned to the participating districts in proportion to the amount contributed by them. Any such board of directors shall have power to give such notice of the landowners’ meeting as it may deem proper, in addition to the notice required by section 245.460.