Black Eyed Susan

I think you'll need to dig them up to get to the root of the problem.
Give them a good wash, strip the affected leaves and then replant in a different locale.
Finally, return to the original site, and pour a full kettle of boiling water into the hole.
Job done!

By Black Eyed Susan, are you referring to Thunbergia alata (the climber), or Rudbeckia? Both have this common name.

I'm growing Thunbergia alata in the conservatory by the window, and after flowering wonderfully, the stems and leaves have all turned white and there's only a handful of flowers on the end of the stems. I've seen some very healthy ones growing outside. Would I be better to plant it out now, or can I cut it right down and expect flowers in the autumn?
(Sorry to 'hijack' your question Cynthia, but I thought the thunbergia experts might already have looked at this question)

I grow Thunbergia gregorii (orange clock vine) as a perennial in a pot and it has never failed me for 3 years now. I leave it in the pot outside and cover the top with a cloche during the colder months. I potted it with a mixture of JI no3 and compost with grit and slow release feed pellets. I never overwater it as I think your problem may be overwatering, I give it a good liquid feed when it starts to show new growth in the Spring. When the growth has finished and the plant is going into hibernation the stems should turn a dry straw texture, cut all of this back, cover the top of the soil with fleece, then bark and then put a cloche over the top if you are leaving it outside.