Pull down resistor value in debouncer circuit
Figure 1. The relevant details from the 74LS datasheet.

Understanding the inputs:


We can see from (1) that to guarantee a low reading on the input we need to pull the input below 0.8 V.
From (2) we can see that the \$ I_{IL} \$ - the current into the input when held at 0.4 V is -0.4 mA. The negative sign tells us that current will flow out of the input into whatever is pulling the input low.
From the line above we can calculate the required pull-down resistance from Ohm's Law as \$ R = \frac {V}{I} = \frac {0.4}{0.4m} = 1k \ \Omega \$. Any higher resistance may not pull the input down adequately. Any lower resistance means unnecessary loading of the previous stage.
From (3) we can see a characteristic of TTL which is that they pull high by themselves - although it's best not to rely on this - and that only 20 µA is required to hold it high.


Understanding the outputs:


From (5) we can see that the \$ V_{OH} \$ - the high level output voltage -
can supply only 400 µA - not enough to power your LED reliably.
Below (5) we can see that the \$ V_{OL} \$ - the low level output voltage - can sink 4 to 8 mA and still stay low enough to drive another TTL chip reliably. This determines the maximum fan-out of the TTL chip - the number of TTL inputs one TTL output can drive.




Figure 2. 7400 NAND gate. Source: Wikipedia's Transistor-transistor logic which gives a detailed explanation of the workings.

To understand why all of the above is so it helps to look at the 74xx internals. (I coulnd't find a 74LSxx output stage diagram so this standard TTL output will have to suffice.)

The inputs:


The inputs are formed by a dual emitter transistor. When disconnected no current flows through the emitters. Pull up requires little or no effort - just enough to make sure that noise or EMF doesn't turn them on.
To pull low the base current has to be "sunk" by the external circuit.


The outputs:

The output is named a "totem-pole" structure after the native American Indian carved wooden totem-pole.


The output pull-up is weak for a few reasons. At (1) we can see that there is a 130 Ω pull-up internally. This is the first cause of voltage drop on the output. We can also see that the high-side transistor switch is NPN so the emitter can only go as high as 0.7 V below the base voltage which itself will be less than 5 V due to R2.
Meanwhile, at (2) we can see a strong pull-down switch formed by the lower NPN transistor.


This explains why the TTL pulls down better than it pulls up.


  But after 15 minutes the output Q becomes active even if R and S are both low and previous output Q was also low.


I suspect the output voltage is dropping with time due to heat. You are probably overloading the output and the output voltage and input threshold voltages are changing over time.

Try increasing your resistors to 1k and disconnect your LEDs. Monitor the output with a voltmeter and see if they remain stable.

To power the LED you can try reversing it to see if pulling low works better. Limit the current to about 5 mA and you should be OK. With 2 V across the LED and 2.5 across the resistor this would be \$ R = \frac {V}{I} = \frac {2.5}{5m} = 0.5k = 500 \Omega \$. 470 or 560 Ω would be fine.