Costs: $7 Battery, $50 to have it replaced
No improvement, if anything worse. Does not respond to wireless charger, and battery still drains fast.
Bought this battery: http://www.amazon.com/gp/product/B00NCJMQXK?psc=1&redirect=true&ref_=od_aui_detailpages00
I had the exact same problem. I can't give you a link to the one I got as it was an NZ retailer, but here's an Amazon link to what's claimed to be an OEM replacement.
The battery model is BL-T9.
I got my battery from Amazon (vendor Fly-lucky) Apparently the battery came from China, It took around 1 week to arrive. The battery seemed legit, look exactly as the original. Unfortunately I didn't write down the manufacturing date. I immediately noticed the life time increase with Lollipop. (from ~5 to ~10 hours of normal use). Now, with Marshmallow, and with the wifi drain solved, the life is spectacular (around 24 hrs. of normal usage, overnight drain of ~3%)