Some circuits oscillate without one.
A battery has an internal resistance that increases as it runs down. Then the supply voltage changes as the load current changes. A supply bypass capacitor smooths the voltage changes so they don't change sharply and the change is much less. The capacitor powers the circuit when the battery can't.
Also how would I go about calculating the needed value. (I'm thinking it has something to do with the current or voltage.) Does this also mean I will need one for the Indicator circuit? Or will this act for both?
If you have only one battery then your circuit needs only one supply bypass capacitor. If it has RF then a ceramic disc capacitor must also be used since it works well at RF frequencies and an electrolytic capacitor works well at low frequencies.
A very low current low frequency circuit needs 0.1uF to 10uF.
A "medium" current low frequency circuit needs 100uF to 1000uf.
A high current low frequency circuit needs a high capacitance.
Also could I not have just adjusted the 560ohm resistor so that the current was still 22ma. I'm not going to but I'm just wondering.
If the 560 is changed to 620 ohms then the LED current in the circuit with the pots is 22mA. The max current for most LEDs is 30mA to 40mA. You won't notice any difference between an LED at 22mA and another at 25mA. Double the current is only a little brighter because your vision's response to brightness is logarithmic (before your iris appies its automatic brightness control).
10uF would probably work fine in your circuit. If it had audio as well then 100uF or 1000uF would be needed to keep the audio from being affected by the LEDs turning on and off.