@JohnSimpson,

Just to make sure, the LED lights that you have connected are capable of dimming, correct? The KP405 is one of the only Kasa Plugs to be capable of delivering a lower power amount depending on the brightness. If the LEDs have a minimum threshold or are not capable of dimming, a fluctuation in the power could cause the lights to shutoff for safety.

For next/first steps; I would recommend resetting the plug using either a soft or factory reset and reconnect to your network; if you think that it is a network setting still; I would try to set an IP reservation on your router for the device. 

When the behavior next occurs, try disconnecting your phone from your home network and then attempt to control the plug via a cellular connection. If it cannot be controlled, it would indicate that the plug has lost its cloud connection.