Why is my iPhone 5s battery suddenly draining so fast? Even after clean reinstall!

I've had a problem with my iPhone 5s battery, which started on 31st October BEFORE I updated iOS to the latest version.

All of a sudden I found my battery draining by 10% an hour, it even drains if all apps are shut down and the phone is in airplane mode. First thing I did was update to the latest ios, no change. I switched off background app refresh, no change. I rreset my phone, no change. I wiped and restored my phone through iTunes, no change. I went to the Apple Store and they did a battery check and my battery is at 103% (ie very good condition) and said to do a "new phone" reset. Now I've done that and no change. It's really frustrating as there appears to be no obvious software or battery problem yet my phone drains away within just a few hours with no use, even if in airplane mode. Note that Facebook isn't installed (suggested to be the cause of a similar problem recently).

My phone is just out of warranty, and Apple say they want £199 to fix the handset (or more accurately, to give me a replacement), which I find frustrating when they can't diagnose any problem - how do I know it won't just happen again?! It's unclear whether this situation is covered by UK consumer law (up to 6 years reasonable expectation to be free from defect http://www.apple.com/uk/legal/statutory-warranty/ ).

Roll back 12 months where similar problems reported on forums were reportedly down to issues with iCloud accounts, and high data usage for "documents and sync" background activity. As it happens my data usage seems unusually high for the last month, but if I turn off iCloud and mobile data there is no significant change in my phone performance.

if my battery is fine, and the proble started overnight with no obvious cause (I don't recall installing any new software just before it happening), what else can I reasonable do to try to fix this problem? It's a trek for me to get to the Genius Bar and, helpful as they were at the time, I feel like they've done all they are willing to do right now.