Certainly nitrite should be zero in a cycled tank, that it isn't shows there's something amiss. Are you measuring with a liquid based test kit?
If your fish are tolerant of salt (honestly I have no idea what a red-tail tiger shovel nose hybrid is!) that would be a good start here as it will work against the ich, the tail rot and the elevated nitrite.
At the same time you want to be keeping the nitrite below 0.25 at all times with water changes (suitably dosed with salt to keep the concentration up) until the filter picks up again.
Don't introduce salt all in one go. Dissolve it in some tank water and add it over the course of a few hours.