No battery rechargeable led light
I have a feeling you want something similar to the way that mobile phone wireless charging works. but instead of it charging a battery in your mobile phone you want to charge a capacitor which can then be used to power your LED.

The following reference is useful to what I believe that you are looking for:
WIRELESS LED CIRCUIT



A notable difference would be, instead of just the inductive coupling powering the led, you wish to store that energy to power the LED for 10 minutes.

You will then find the following question useful for calculation the discharge time of the capacitor:
CAPACITOR DISCHARGE CALCULATIONS

However, after the above being said it wasn't made clear why there is a reason to use a capacitor and rely on keep charging it. some batteries are designed to last years especially at such low current draw.

EDIT: 
The following link regarding capacitors being used to power and LED may be of some use to you: CAPACITOR LED