Okay, it’s been coming up to two weeks since you posted your question and nobody has answered (not at all surprising on this forum, I must say). I’ve been waiting to see if anybody would before venturing an opinion.
So just now, I looked up undecylenic acid on the May Clinic website. From what I can make out, it is an effective fungus killer, but it is used externally. Personally, I would be extremely hesitant about using it in the windway where it could come in contact with moisture which I would be sucking back up into my mouth.
Vinegar is also an acid, and as such, should kill off the mildew it comes into contact with. And the hydrogen peroxide is a bleach that should render the (hopefully) dead mildew colorless. Both are safe to get into the mouth in my opinion.
I’m not familiar with thymol. Wikipedia says that it may be a fungicide. A google search indicates that it is toxic. Based on that, I would hesitate to use it for this application.
Looking at your attached photo, I suspect that what treatment you have already done has worked on the surface and slightly below. What remains appears to be discoloration and little else. I also suspect that the mildew will not come back.
However, that can be easily checked by playing the recorder for a week or two, blowing out the condensation and swabbing, letting it dry between playing sessions, and periodically popping the block out to check whether the mildew has come back.
If it doesn’t return, you have a recorder. If it does return, you have a nice piece of firewood. Alternately, you can pop the block, paint the windway with Kilz, get a lamp kit from your local hardware store, and make an attractive lamp for your music room.
All the best.