The large surface area and rich functional groups of a 2D nanostructure, sulfonated graphenal polymer (SGP), provide a new strategy to assist the dispersion of carbon nanotubes (CNTs), far better than the dispersing ability of graphene oxide and sodium dodecyl benzene sulfonate. The efficient codispersion of CNTs and SGP facilitate the fabrication of composite buckypapers with wide-range tunable fractions of CNT, SGP, and polymer, like poly(vinyl alcohol) (PVA), making it possible to search out the most optimal structure in the fraction space of the constituents. The globally strongest buckypaper is obtained at a CNT:SGP:PVA mass ratio of 7:3:10. Owing to the super hydrophilicity of SGP, and the hydrophilic characteristics of PVA as well, the composite structure can be reassembled with the aid of water, resulting in easy removal of creases and efficient cut heal/repair to broken papers. The healed papers can exhibit about 80% recovery of the tensile strength. Furthermore, just by mechanical stirring, the composite buckypaper can be redissolved in water for a totally green recycling reuse.