The K{sup -}p{yields}{pi}{sup 0}{pi}{sup 0}{sigma}{sup 0} reaction is studied within a chiral unitary model. The distribution of {pi}{sup 0}{sigma}{sup 0} states forming the {lambda}(1405) shows, in agreement with a recent experiment, a peak at 1420 MeV and a relatively narrow width of {gamma}=38 MeV. The mechanism for the reaction is largely dominated by the emission of a {pi}{sup 0} prior to the K{sup -}p interaction leading to the {lambda}(1405). This ensures the coupling of the {lambda}(1405) to the K{sup -}p channel, thus maximizing the contribution of the second state found in chiral unitary theories, which is narrow and of higher energy than the nominal {lambda}(1405). This is unlike the {pi}{sup -}p{yields}K{sup 0}{pi}{sigma} reaction, which gives more weight to the pole at lower energy and with a larger width. The data of these two experiments, together with the present theoretical analysis, provide firm evidence of the two-pole structure of the {lambda}(1405)