I see no reason to bleach anything and start over. If you're sure the tank is cycled (meaning 0 ammonia, 0 nitrites, and measurable nitrates), you could probably just leave it running, perhaps adding a teaspoon of pure ammonia right before you leave.
Most fish can live with parasites for a while at least before killing them. I think they would likely display other odd behaviors in the meantime, like scratching, not eating, or have worms hanging out of their anus, so the quick deaths in succession does seem to point to the water, but without more specifics it's hard to say.
edit: just saw your pics. Hard to tell for sure, but it's possibly Columnaris, which would indeed kill them quickly. If that's the case, disregard what I said previously and bleach the tank.