ammonia can go undetected on a hobby test kit as it measure levels safe for fish (0ppm) so that still could be an issue. PO4 is blamed for all algae species, it isn't true. If it was then somebody explain to me why people can dose PO4 without getting algae?
Does the tank get any sunlight on it?
Get some fast growing weeds such as egeria densa (pond weed) and hygrophillia species. Moss balls are slow growers so they wont use many nutrients.
algae blooms can be hard to get rid off, so plants will take a long time although they will help in the future.
Large water changes do not seem to always help, but you could try doing an 80% water change, followed by a 3 day blackout, followed by another 80% water change. The large water changes will help to remove the majority of the algae spores.
A UV steriliser will clear it up very quickly but it can be expensive.