Just to add my 2p worth.
I turned the light on the phone on and kept doing video recording to drain the battery, I let it die completely, recharged it and the wireless came back on. Powering on and off through the software didn't work. None of the other suggestions I'd read were any help, and to be honest some were stupid.
The best thing I can think of is that there is some hardware on the wifi that gets put in an inconsistent state and that powering on and off does not fix it as there could be some sort of trickle going through the hardware to keep it alive. Draining the battery allows the hardware to be really turned off and it puts it back into the right state.
No idea how long this will last and no idea what caused it, it wasn't due to an IOS upograde as I did that weeks ago.
Anyway, it would be nice to hear from Apple about this issue. Though given their appalling track record of acknowledging any faults I doubt we'll ever get a response.
If this hadn't worked I'd have cut my losses and dumped my iPhone and moved to a Samsung, if I get any more hassle with the iPhone I will!
I've been buying Apple since the Apple Lisa in 1984 and currently have five Mac's, three iPads, n iPods and 3 iPhones at home. The lack of innovation in recent years (don't tell me that a super thin iMac or a high res Macbook is innovation), the constant legal challenges from Apple, the poor QC that appears to be coming from Apple now (and this started on Jobs watch, not Tim Cooks) makes me think Apple may be losing its way. I know the share price is super high, but I'm old enough to remember IBM ruling the roost, Microsoft being number 1, Cisco being the most valuable company, Sun being a force to be reckoned with. As with the Roman empire, the British Empire, the Soviet Union, things do come to an end and things do change. To quote Churchill
"Now this is not the end. It is not even the beginning of the end. But it is, perhaps, the end of the beginning."