I hope you don't mind the question. Is this your parents tank?
A tank being cycled means that the filter has gone through a process that allows good bacteria to grow in the filter to process the ammonia that is produced by the fish and transform it into something less toxic. If the filter has been running for years and the filter media (sponge, floss, whatever is inside) has remained... then your tank is cycled.
It would be best to start regular water changes for the health of the fish and to keep the nitrates at a safe level as well.
Sorry to not be much help for your fish but if this is the case with your tank ... that it hasn't been properly maintained, the water is probably what is making your fish sick.