Ned help modifying a Harbor Freight LED flashlight into a light source for a microscope
Short answer: 18 ohms.
Your USB may be one that complies with the spec that limits current to 100 mA. If your LED circuit wants more, it may be cut off entirely. While USB may provide 500 mA, you must request the extra current via enumeration, which involves USB message transactions (something you don't want to get involved with). So to be safe, a 18 ohm quarter-watt resistor should limit current to your 14 parallel LEDs to 100 mA.

Tony Stewart provides a similar approach, suggesting a 6.4 ohm (one watt) resistor, which would provide about 280 mA. Your USB port may not comply with USB specifications and provide any current in which case, you can get more light with his method. But your USB source may balk and shut down because the load is beyond the 100 mA initial limit.

However, you have not said how much current your microscope-to-USB interface is flowing. Your extra LED illuminator will add to this current.