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How do I regulate voltage AND current?

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I'm a newbie, and rather than experimenting with beginner projects which serve no practical purpose once created, I'm trying to work on my own small projects which can be used for something afterwards.

For one of my first projects, I wanted to design what is basically a deluxe bleeder jumper for draining capacitors.  Into a small enclosure, I'm going to use a potentiometer so that I can adjust the resistance and therefore the bleed time for any given capacitor.  This way, I don't have to take 10 seconds for one and 60 seconds for another. 

In addition, I wanted to set up an LED so that I would know when the capacitor was fully drained (I figure if there's not enough juice to light an LED, I'm safe).  But here's where I run into my first obstacle.  There is going to be a large variation in voltage and current going into the circuit - so what is the best way for me to regulate a fixed maximum voltage AND current going to the LED?  (I figure it's good to end up with a max of 2 volts at 30 mA)

Doing some preliminary research, I see that there are Low Dropout Regulators which I can use to regulate voltage (I'm not saying I'm sure I know how to use them, but at least I found them!  :) ), but even those allow output currents that would fry an LED.  So, what is the best way to handle this?

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Make a circuit that will drain the capacitor. The pot can control how much current a transistor can drain and therefore how long it takes.

The LED will also drain the capacitor so it should have its own power supply. A circuit can be made with a comparator to turn on the LED when the voltage across the capacitor is low enough.

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I apologize for being unclear.  My original idea was not to have the LED light up when the capacitor was empty, but to have it be powered by the capacitor being drained.  Thus, what I would expect to see is the LED lighting up and then slowly dimming until it was off when there was no longer enough charge to sustain it.

That way, I would have a real-time indicator of the rate of drain, and this would be a completely passive device, not requiring any additional power whatsoever.  If this is a viable plan, then the only issue is how to regulate the maximum voltage and current so that the charge from the capacitor doesn't fry the LED.

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