There are two possibilities:
1) LEDs can light with a very small current and solid state relays leak a considerable current, even when off.
2) The SSR has failed short circuit, which is the most common failure mode.
Test the solid state relay with an incandescent lamp. If it works, then it's #1 and the solution will be to connect a resistor across the LEDs. If it still doesn't work, then the solid state relay is bad and needs replacing.