What is causing the noise is the rapid change in current demand. Capacitors can react to these changes faster than batteries and power supplies so they will "absorb" the spikes. The power supply can't react to the change in current demand as quickly as the capacitor can so I wouldn't worry too much about what "source" the LED is drawing from if both the cap and power supply are there. I would just stick a capacitor across the power rails so that the capacitor will "absorb" the spikes and reduce noise. In this way, the capacitor should charge up faster than it can discharge too (since there is a resistor in the discharge path, while there is nothing in the charge path).
With how you have it right now, I don't get why you need the 10ohm resistor- all it would do is sap power and make the LED slightly dimmer. I also don't see how it would make that the LED would take power from the capacitor...there is already R1 sitting between the LED and power supply and a short between the cap and LED. The cap is already an easier path. R1 also limits how quickly the capacitor can charge up. It's just not needed.
Size also doesn't matter too much. The LED is probably being switched at low frequency (not MHZ, not even kHz. 30Hz-60Hz is all that is needed to fool the human eye). So you just use a "large" capacitor. 47uF is probably fine, as is 10uF or whatever else.