Also you might want to check your serial number on Apple's website.
https://www.apple.com/ca/support/iphone5-battery/
They are (or were, not sure if it is still valid) free replacement batteries for certain iphone 5 serial numbers.
Quote:
Apple has determined that a very small percentage of iPhone 5 devices may suddenly experience shorter battery life or need to be charged more frequently. The affected iPhone 5 devices were sold between September 2012 and January 2013 and fall within a limited serial number range.
If your iPhone 5 is experiencing these symptoms and meets the eligibility requirements noted below, Apple will replace your iPhone 5 battery, free of charge.
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