Check your Power button with an ohm meter if you have one, with computer unplugged and it's wires removed from motherboard header " power switch wires".
You will need 3 hands to do this, one lead on one of the wires, other lead on the remaining wire. With a third hand push the power button. Do the ohms on your meter start to climb. If so your switch works.
You can also jumper on mother board header with a small flathead screw driver the power switch pins. Just watch what your doing and have good light.
Also make sure your new battery is touching prongs all the way.
if this all turn out ok, like already suggested move on to power supply.