When you are attempting to start the computer, are you getting no power to the computer... or are you just getting chimes?
Either way, it is likely that your battery is placing unnecessary pressure onto the ribbons that connect to the bottom of your upper case. That ribbon runs information for several things including your keyboard, trackpad, and power button.
Ok, so let's find out if it's something other than battery swell.
Start you MBP up without the battery in it, power adapter connected of course.
After start up, place the battery back into its proper place in the battery bay.
If the machine functions with the battery in, select "restart" from the Apple Menu in the top left bar of your screen.
If that does not work or the MBP did not function properly with the battery in try these two steps.
1) Reset your NVRAM (is still often referred to as PRAM even though that was for Power Books and before) * You can leave the battery out for these.*
- Start up your computer by pressing the power button, and immediately holding down the Command + Option + P + R keys simultaneously. HOLD them.
- Continue to hold them through the start up chime. The screen will go black again and you will hear the sound of the drive again... then a second chime.
- As soon as you hear the second chime, release all of the keys at the same time.
After that is complete, you can try the battery. If it is still not functioning properly, follow these steps.
2) Resetting the SMC (System Management Controller) - This should not be done first without having tried other options such as restarting your computer, NVRAM resets, etc.
- Shut down the computer.
- Disconnect the MagSafe power adapter from the computer, if it's connected.
- Remove the battery.
- Press and hold the power button for 5 seconds.
- Release the power button.
- Reconnect the battery and MagSafe power adapter.
- Press the power button to turn on the computer.
If none of this works, I will have to ask this. Before the replacement of the logic board, did you have the same battery issues? If not, how was the battery stored since it's last use, and has it been dropped? If it has been dropped or exposed to highly varying temps, then your battery most likely has some swelling and is providing too much pressure against the bottom of the upper case.