I've used original Dawn dishwashing liquid for years with no harm to my daylilies. I've read that Ivory dishwashing liquid is used with equally good results. I live on the Louisiana gulf coast so heat is an issue 8-9 months out of the year. When I spray, it's early of a morning before the sun can create a prism effect on wet leaves. I feel if soap/detergents have to be washed off after being sprayed, too much was used in the first place.
I have never had spider mites (touch wood), but have done a lot of reading about them. Nasty critters.