The recently published experimental data on K+Λ photoproduction by the SAPHIR, CLAS, and LEPS Collaborations are analyzed by means of a multipole approach. For this purpose the background amplitudes are constructed from appropriate Feynman diagrams in a gauge-invariant and crossing-symmetric fashion. The results of our calculation emphasize the lack of mutual consistency between the SAPHIR and CLAS data previously found by several independent research groups, whereas the LEPS data are found to be more consistent with those of CLAS. The use of SAPHIR and CLAS data, individually or simultaneously, leads to quite different resonance parameters that, therefore, could lead to different conclusions on "missing resonances." Fitting to the SAPHIR and LEPS data simultaneously indicates that the S11(1650),P13(1720),D13(1700), D13(2080),F15(1680), and F15(2000) resonances are required, whereas fitting to the combination of CLAS and LEPS data leads alternatively to the P13(1900),D13(2080),D15(1675),F15(1680), and F17(1990) resonances. Although yielding different results in most cases, both SAPHIR and CLAS data indicate that the second peak in the cross sections at W∼1900 MeV originates from the D13(2080) resonance with a mass between 1911 and 1936 MeV. Furthermore, in contrast to the results of currently available models and the Table of Particle Properties, both data sets do not exhibit the need for a P11(1710) resonance. The few data points available for target asymmetry cannot be described by the models proposed in the present work.