With Mac unused and unplugged for a full month, what might have happened is that battery reached its heaven and went bad and unable to take charge any longer. First step would be to try and disconnect the battery and see if that gives any change and Mac would be able to start on its own when plugging in the magsafe. If it starts without the battery then a battery replacement may probably solve all issues, otherwise we get into machine hardware failure that would need more or less blindfolded parts replacements to progress any further with troubleshooting.