How to build a small LED assembly?
A Christmas string LED is not the best choice. Digikey has thousands of different LEDs and switches. Learn to use the filters. Avoid surface mount parts (harder for a beginner). 

For LEDs, you probably want through hole parts. A common size is about 0.2" dia.

For the switch, you probably want slide, panel mount. Here is one possibility:
https://www.digikey.com/product-detail/en/cw-industries/GF-126-0135/CWI437-ND

An appropriate battery holder for a coin cell may be the hardest item to find. Most are designed to be soldered to a printed wiring board and the board helps contain the battery. 

I have keychain LED lights that don't use resistors, but unless you get a good match between the battery voltage and the LED voltage, it is best to use a resistor. LEDs are highly non-linear, the current is not proportional to the voltage, and the current is what you want to control.

It is hard to pick a resistor value without knowing how may volts your battery will be, and which LED you are using (different colors drop different voltages). You want a separate series resistor for each LED, LEDs in parallel do not share current well. A common LED is rated for 10-20 milliamps, but will light fairly well with only 1-2 milliamps. You want to be on the low-end so your battery will last.



http://lednique.com/current-voltage-relationships/iv-curves/

Pick your desired current on the left, and draw a horizontal line until it intersects the LED curve. Then draw a vertical line and see what the voltage drop will be. At 5 mA, white LEDs drop about 2.4V and red LEDs drop about 1.6V. Use the curve supplied with the actual part you have chosen if possible, but this generic curve will get you close.

Series_Resistor = (Battery_Voltage - LED_Voltage_Drop) / Desired_Current







simulate this circuit &ndash; Schematic created using CircuitLab