It sounds like your iPhone 5 is not charging. I found an article with some steps that might help with this. I recommend reviewing the steps in the sections titled "Battery does not charge from the USB power adapter" and "Will not turn on, will not turn on unless connected to power, or unexpected power off" in the following article:
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Thank you for the additional information. Due to the nature of your issue or question you may find more information by using another one of Apple's support resources - https://expresslane.apple.com/GetproductgroupList.action
All the best,