Make sure you're trying to plug it in the right way. Some plug need to snap in to be seated - try doing so but without forcing it. Frankly, if it were me, I'd contact eMachines support
here for help with this.
If I were support, first question I'd ask is: how did the power button cord fall out? I'm assuming you had the machine open - what were you trying to do? If there was some other problem or you were installing hardware in your machine, its not starting may have nothing to do with the power plug.