You may have some misunderstanding here.
The Arduino is not in any respect whatsoever, a power supply. It does not "power" anything.
To power 5 V WS2812 ("NeoPixel)" LED strips you use a 5 V power supply. For up to 2.1 Amps, a USB "Phone Charger" is often a very convenient and inexpensive supply for up to 35 WS2812 LEDs as well as your Arduino.
You will need to connect the 5 V directly to your LEDs and also to the Arduino. You can connect 5 V to the "5V" pin on the Leonardo except when you have it connected to the PC via USB. You could connect the 5 V to the Arduino via the USB connector but you must connect the 5 V separately to the LED strip. 