SSR staying on (not reacting anymore)
The SSR has a triac as switching element. A triac needs a pulse on its gate to go on, but after it goes on it will stay on until the load's current drops below a hold current. This makes that a triac isn't suited for switching DC, only AC, because there the current will drop below the hold current twice per mains cycle. 

So don't use it to switch the LED, that's DC. Besides, you don't need a relay at all (not even a mechanical one) to switch a LED on and off. For that we usually would use a transistor.