One way to do it is put a diode in series with a smaller valued resistor and then put this diode/resistor combination in parallel with the 100K resistor so that the diode is reverse biased on charge and forward biased on discharge. That way when in the charge mode the diode is reverse biased and so only the 100K resistor is determining the rate of charge. Now when the discharge mode is selected the diode becomes forward biased and now the second resistor is in parallel with the 100K and so the discharge rate is speeded up.
I accidentally reversed the sense of forward-bias and reverse-bias in my recommendation so I corrected it in my earlier reply. I just wanted to let you know.
Is there a formula to determine the discharge rate??
and how much would the LED affect the discharge time?
how can you make charge/discharge the same time??