Sabrina, there are something like 7,000 different plant rusts, and most are specific to certain plants. Only daylilies and Patrinia can be infected by daylily rust (although there has been a case in another plant in the daylily family, Dianella).
Your pictures do not look like daylily rust at all. Check out this FAQ page, scroll down to the images for differentiating:
http://web.ncf.ca/ah748/FAQ.ht...
Yes diluted dish soap as a leaf spray can work for daylily rust, this was shown in a scientific study. Not as effective as a systemic fungicide but still worked to some extent. I believe it was probably lemon "flavour" which is what is preferred for insects. You can also buy fungicidal soap products for plants.
Daylily rust spores do not primarily live in the soil. They could get washed off a plant by rain or irrigation and then splashed back up. Primarily they are wind blown. But you don't have daylily rust
