This is what I would do... at the very least it won't work and you've only lost a few hours. Wind your balls into skeins (if you don't have a swift, someone's arms or the back of a chair will work), and tie the skeins loosely with some cotton yarn/shoelace/etc--don't use wool, it might color your yarn. Then fill up the sink with luke warm water and let it soack for a while. Hang up the skeins--you can weight them down with a can of soup or whatever so they don't end up kinky--and when they're dry check them out; if the hair is gone, great, if not, well at least you have an excuse for yarn shopping!
I would do the white separately from the other colors, just in case they bleed.
Also, if you still have the yarn bands, call your LYS and ask them to put aside some of the same dyelot for you: that way if you DO have to get more yarn, it will match.