Are the white patches still there? I think the thing we need to know is the nature of the patch.... i.e., Brooklynella looks like a white or slimy patch attached to the fish on top of its scales. A "scrape" will look more like the scales missing, and ich is a well defined sugar-sized speck. Ich doesn't accumulate in patches, while you will often find it on the pectoral or other fins first, I would not use the word "patch" to describe it.
IME, neon gobies are notoriously ich-prone, and the presence on this fish just after adding it to a new tank may have nothing to do with what you saw previously on the angel.
Tom is right, a QT is the best way to deal with ich. Moreover, a QT is the best way to avoid ich.
That said, the cleaner shrimp. UV, and garlic approach he describes is quite effective in treating (more so preventing) ich in a very well established, otherwise very healthy system. If your fish are primarily new additions, this might not describe your system.
Hope that helps,
Abby