Hi, 415.
Does not run with battery out and plugged in? Correct?
Unplug, remove battery, and hold down power button 30 seconds.
Re-install battery, plug in and see if it will run, or charge.
(This "Reset" sometimes helps.)
New jack may be faulty, when I install a new jack, I always
test before reassembly, by checking that voltage gets to motherboard.
Damaged old jack may have caused a problem on the motherboard.
Sounds like the service center really let you down.