Vbe for a single transistor is typically one diode drop's worth of voltage, which is where the 0.7V comes from.
TIP120 however is 2 transistors in a Darlington configuration with one transistor driving a 2nd transistor so that the source can supply less current and still achieve higher gain.
Since you are intending the TIP120 to be used as a switch, you will drive it hard enough to ensure it goes fully into saturation when turned on.
From Figure 2, the lowest Vbe(sat) is 1.25V.
So base resistor of (5v - 1.25V)/20mA = 187.5 ohm, so use a 180 ohm resistor.
Actual collector current flow will then be determined by the current limit resistor used with the LEDs, typically seleceted for a max continuous current of 20mA max.