Efficiency roll-off is a major issue for most types of light-emitting diodes (LEDs), and its origins remain controversial. Here we present investigations of the efficiency roll-off in perovskite LEDs based on two-dimensional layered perovskites. By simultaneously measuring electroluminescence and photoluminescence on a working device, supported by transient photoluminescence decay measurements, we conclude that the efficiency roll-off in perovskite LEDs is mainly due to luminescence quenching which is likely caused by non-radiative Auger recombination. This detrimental effect can be suppressed by increasing the width of quantum wells, which can be easily realized in the layered perovskites by tuning the ratio of large and small organic cations in the precursor solution. This approach leads to the realization of a perovskite LED with a record external quantum efficiency of 12.7%, and the efficiency remains to be high, at approximately 10%, under a high current density of 500 mA cm-2.