How do I figure which LED I need?
Figure 1. Typical IV curves for various colours of LEDs. Image source: LEDnique.

Replacement of any of those small indicator LEDs with 3 or 5 mm LEDs should be fine.

A little bit of background theory may help. LEDs have a non-linear relationship between applied voltage and current. The forward voltage also depends on the colour as shown in Figure 1. 

Your measured 3 V LED voltage suggests that you've got either white or blue LEDs on the board. With the 1 k&Omega; resistor in series the current will be limited to a safe value even if you change to one with a lower forward voltage.