Your powdery mildew will not kill the rhizomes, you will see more new growth in the spring. Just for future reference powdery mildew development progresses due to peroids of cool damp weather followed by dry warm weather. You can use some products to curtail the activity of the fungus. One that comes to mind would be Copper sulfate, I have also recently tried out a product called Messenger. It is a great product that "tricks' the plants immune system into thinking it is being attacked. For further information and availabilty go to the website http://www.edenbio.com/usa/agriproducts/?m=1
this product is a harpin protein that increases the plants ability to deal with attacks by pests and pathogens. It is totally safe and can be used on edible crops without worry. If you need any further info or advice let me know.