apple logo loop ...

Seems to be a common problem I suspect prying damage on the motherboard did u pry on the motherboard side at all when taking battery out

Probably the battery.

charge your new battery in the known good iPhone 5 with the known good logic board. Switch the 'bad' board into this phone and see if it works. if no--pry damage if yes--bad 'new' battery.

jessa

The four screws that hold the metal cover over the group of plugs from the touch screen digitizer must be returned to their original positions or "fatal" damage will occur in the holes @ 4 o'clock and 7 o'clock (lower left and right phone on its back charging port towards you). If you look down the holes there are 5 or 6 traces that run directly underneath the screw. The screws are different in length by just a hair, but it's enough sever the traces running directly underneath. When I did it, the phone would still turn on, but was stuck in loop similar to what you're describing. use a scope and check both of those locations. if you dont have a microscope, a pair of binoculars flipped around will work.