When I shine a laser into the water I see a few large sparkly things in the beam. Are those big chunks of silver?

No..well maybe some silver plateout that fell off an electrode.. Mostly, it's dust, pollen, fiber particles or possibly chucks of silver oxide that got washed off the blackened electrode somehow and hydrogen bubble/silver "structures" that vanish in a few days The air is full of particulate matter. Check your local sunbeam. Wiping anything leaves fibers etc. behind.

Filtering removes it pretty well. Or, let it settle and 'decant', leaving the stuff in the bottom. [That's what works the best]

It's harmless anyhow.

A coffee filter is sufficient. Rinse it with distilled water before using in order to remove any chemicals that might be in it.