How to make auto off functionality for an led lamp circuit with only transistor, capacitors, timer 555 chip?
In your circuit you have posted, remove R7, replace it with a 680uF capacitor. And then, connect the base of that transistor to the ground/emitter using a 1Megaohm resistor. 

This way transistor will get the base drive through the cap for about 10 minutes until it is fully charged by the 1 megaohm resistor. This mod is the easiest one to accomplish your objective. Of course more can be done to reset the timing cycle etc by adding more components. 

555 is not really needed unless you want to have more precise control of the timing.