Never dealt with brook, but I've seen ich, and kind of looks like it to me. As I too have ich in my display (not active), I've read up on the subject of eradication. I've come to the conclusion that without drastic measures it's nearly impossible to rid your tank of it. Fallow periods often fail because ich can lay dormant for far longer than the usual fallow period in anaerobic areas of the tank, such as in the sand bed. I'm not saying don't fallow the tank, but you may want to consider a longer period, 80+ days.
Depending on your setup, I have a couple suggestions. Instead of quarantining your fish, if possible remove the corals and inverts and use hyposalinity on the display. This should, if done properly, kill the most determined ich in your tank. Then the inverts/corals can sit out the fallow as they may be carrying encysted ich. Once they are done the fallow, give each one a thorough series of rinsing to wash off any potential surviving free swimmers.
Quarantine your fish, treat them for the ich until it goes away, then return them to the display and manage the ich. Feed your fish a quality, vitamin supplemented diet and keep stress to a minimum. Typically they can develop a resistance to the ich, and can fight it off as long as they stay healthy. Unfortunately this is really a bandaid solution, as one thing falls out of balance and you'll get another break out.
