If you use a known good battery and still get this fast discharge, then consider the following:
First possibility is that you a have a short somewhere on the board, causing battery to be depleted because this short is consuming power and dissipating heat.
And also the charging circuit in the iPhone 5 is infamous because of a component on the board known as U2 which often fails. The risk of breakdown increases with the use of bad quality charging cables and car or wall chargers.
There are 2 major symptoms that I know of, that indicate the failure of this component:
A/ The phone will charge *seemingly* normally, but if ever it shuts off, it won't turn back on. Change the battery, and it will work normally until again it shuts off. Then again it won't restart. Chronic.
B/ The battery will discharge very quickly and will be dead within 1-2 hours!
C/ Device will not communicate with computer via USB. As if never plugged. And/or you get fake charging (says it is charging but does not).
I don't know how much it costs to fix this in Europe or the USA, but it should be pricey because of the required special tools, skills, and experience.
My advice is to take it to a reputed and trustworthy repair shop that can test the device, confirm the prognostic (may be you're lucky and it is something different and easier to fix), and quote you the repair.
OR have it exchanged at the Apple Store by paying an exchange fee, if still possible (as you already worked on it, they may refuse to do it).