Using a soldering iron apply heat to the connection point where the LED connects to the PCB.

When the solder is heated sufficiently (turns liquid and shiny) insert the LED lead with tweezers. If necessary add additional solder to ensure a solid electrical connection.

When connecting the new LED attach the longer lead to the positive connection point (denoted on the PCB by '+'), and the shorter lead to the negative (denoted by '-').

Repeat for the other connection point.

If the LED is connected backwards the LED will not work.