Temporary, high-power energy storage
A 2.2F capacitor charged to 5V provides an energy storage of up to 27.5 joules: -

Energy stored limit = \$\dfrac{C\cdot V^2}{2}\$ = \$\dfrac{2.2\cdot 5^2}{2}\$ = 27.5 joules.



Clearly you'd need a lot of effort to get it charged up but I don't see why this would not work as a basic energy storage system. Once liberated of it's 10 joules (as per the question), the terminal voltage will be about 3.99 volts and hopefully sufficient to have powered the LEDs down to their limiting voltage but as the question doesn't contain this data I can't be sure.

Also, for the item shown above note that it has an ESR of 35 ohm and this may become a problem in utilizing the energy at a high rate because of volt-drop. For instance, a 1 watt output at 5 volts requires 200 mA and this current through 35 ohms is untenable with this device. You'd need to look for a much better device - something that is a couple of ohms maximum.

If this spec of supercap cannot be achieved then other means has to be found.

EDIT - following valiant work by @helloworld22, it looks like a suitable cap has been found. Here is the data sheet and note that it is on page 10 near the bottom - it is an EMHSR-0002C5-005R0 and it is a 2.5 farad, 5V device with an ESR less than 53 milli ohms. At 200 mA, the volt drop will be 11 mV and negligible and over ten seconds it will waste 21 milli joules to power the LEDs. Nice find!