As Wawa mentioned, you will not get a zero resistance across an SSR when it is on. More importantly, you're not measuring resistance on that meter setting, you're measuring voltage drop as the "diode" test function being used does not measure resistance.The reading of 682 on the meter diode setting would indicate the SSR is switching on, that's the voltage drop across the triac, 682 millivolts is what you'd see for a typical silicon device.You stated you where trying to switch 25 watt led's on and off but you never stated what their rated voltage was. If they are not rated at 100-240 vac, you cannot switch them with that SSR.Nothing is adding up here. Have you actually trying switching a proper load on and off with the SSR?
PS: I have no idea how you're making the measurement that displays the 682 as the black meter probe is nowhere to be seen. Just another mystery that may never be answered.