I'm experiencing the very same issue, albeit on two iPads. A similar - but far shorter! - thread is at https://discussions.apple.com/thread/6166138 and I'll link to that so we all have as much information as possible.
I've updated to 7.1.1 on iPhone and on the two iPads. All working perfectly fine unti then, although it may well be a coincidence. The iPhone is working normally with none of these symptoms, but both iPads (iPad mini and iPad 2) are affected in the same way.
I have experienced the same symptoms as you, namely the available storage will gradually diminish over an approximate 72 hour period to zero KBs available. I have verified that this is not an erroneous warning or reading, in that trying to download an app will fail, as there is not sufficient storage.
That is not to say, of course, that the device should be stating that there is no, or limited, storage available. I am using the devices normally and the total storage used is never more than 5 or 6 GBs. It is totally wrong for it to record that more storage has been used.
I too have found that switching the device on and off again resets the storage reafings, but as you too have experienced the problem then swiftly returns. I have tried restoring the device, both from my iTunes and iCloud backups, and as a new iPad for the reason that the problem could have been with corrupted data in my backup, but the problem swiftly returns.
I took the iPad mini to my local Genius Bar and they were not able to resolve it. They put me on to Apple Support who have been helpful but have not been able to identify the problem. I'm rather dismayed that the possibility of a bug in the release was not mooted by them. Having spent an inordinate amount of time on it - quite beyond what a consumer should have to do, frankly - I returned the iPad mini to the retailer under extended warranty. I am however very sceptical of this being a hardware failure, for instance some issue with the local flash storage, because of the unlikeliness of that, the fact that others including you are sadly experiencing the same issue, and also because there is a similar problem with my iPad 2.
It is helpful to note here the potential issue with iTunes match, to which I too subscribe. I sometimes stream from that (did they at some point change it so it streams rather than downloads on iOs, as with Mac OS X?) but there's no way that it would buffer GBs worth of music. Clearly however it may not be the content of the music but instead some clash between Match and storage.
It would be great if we could all identify this as a flaw or bug in 7.1.1. I can make do with power cycling my devices until it is fixed, but I think we all need Apple to acknowledge and then deal with this as appropriate. If it is not an issue with 7.1.1, then I am running out of ideas.
Best of luck with your issues; if I hear more, and when I get my device replaced or returned, I will post.
