Xperia Z2 :: Phone Shuts Down Even Though It Is Charged

I've had a Sony ZL2 SOL25 for just over a year now. Everything was fine until a few weeks ago. I upgraded to Android 4.4.2 around the same time, but this might be a coincidence.
 
My phone now lasts a couple of hours before shutting itself off, without warning and when there is (supposedly) charge in the battery. It also drops % randomly. From 70% to 55% over the space of 10 minutes. The maximum usage I've been able to get of the phone is 6 hours (about 60-90 minutes use, the rest it's idle). I'm not sure where the problem is: the battery, the battery information on screen, the update?
 
I have factory reset it twice over the last week and if anything, it's just gotten worse. I have added no apps since the last reset, not even configurated email. The problem is not only the very limited battery life, it's that the battery reading is unrealiable, I feel like I never really know what the real capacity is and when it's going to die. What I have figured out is that once it hits 50% that signals the end is near. I rarely see it go under 40%, it will usually shut down before that.
 
More details: This happens at any charge level, for example today it shutdown at 78%. It seemed the battery was having a good day, as I had left the house 2 hours prior and disconnected the charger at 100%, so it had "only" lost 20% during two hours (mostly idle, but about 15min with location services and 20min with facebook/browsing). I got my phone out to take a photo and noted the charge (78%) and right after taking the photo the phone just stated "Shutting Down" and the battery signal in the corner was red (couldn't read %). It would not restart, I tried several times but it wouldn't complete the start-up sequence, turning itself off in about 2-5 seconds. About 30 minutes later I managed to turn it on, it showed the charge as being 16%. I thought I had enough power to send an sms, but as it was sending it shut down again as the battery was 0% and so it turned itself off (this process between turning on and the power showing 16% and shutting down took less than 2 minutes, but the sms went through! Yay for silver linings!). About an hour later I had access to my charger and connected to the power cable, which after 10 minutes said it was at 60% charge.