The keyboard works when on AC only but not when on AC with the battery in, is this correct?
Also with the battery plugged in, the laptop starts straight away as soon as you plug in the AC adapter but it needs the Power button to be operated to start when the AC adapter is not plugged in, is this correct?
Try resetting the BIOS to default values in case the BIOS is corrupted somehow and check the result.
Not sure given what you said, but it may depend on whether you have the battery connected or not, at the time as how to do this.
If the battery is not already connected, turn off the laptop if it is on and when it has completely shutdown disconnect the AC adapter from the laptop. Press and hold the Power button for 30 seconds to dissipate any residual power from the motherboard and then release the button.
Reconnect the battery and then reconnect the AC adapter to the laptop. Hopefully the laptop won't turn on automatically and will only show that it is charging. If this is the situation, then be patient and allow the battery to fully charge before attempting to turn on the laptop in the normal way. If it turns on as soon as you connect the adapter see below.
If the battery is already connected or if it turns on straight away again, (see above) turn off the laptop and when it has completely shutdown remove the AC adapter and disconnect the battery from the motherboard. Reconnect the AC adapter only and then start the laptop by pressing the Novo key and selecting the BIOS option. When in BIOS usually there is a key that is pressed to load the BIOS to default values. Press that, save the changes and exit BIOS and shutdown the laptop. When it has completely shutdown disconnect the AC adapter and reinsert the battery. Try to start the laptop on the battery only and check what happens.