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I had this problem with my iPhone 4S and I tied everything, even two restores. The new iOS burned out the wifi thing. I had to get a replacement phone from my nearest Apple store. My phone is still in warranty, but if yours is not, you have to pay the price you bought it for. I bought my iPhone 4S when it first came out so I would have to pay $199 for my 16GB. If you do end up bringing it to an Apple store, make sure your stuff is backed up to iCloud! I made the mistake of only backing it up to my computer and then when I got there, the guy advised me not to restore from the old update because of corrupt data.
if you need a quick fix before getting it checked out, you can take it out of the case and stick it in the freezer for a couple of minutes. As soon as the phone heats up, the wifi grays out again.
Hope this helped!
What Apple tells you to do on that link doesn't work. You need a replacement.