I suspect you removed the rear case and forgot to carefully remove the connector from the switch.
Most eMac's have been disposed of by now, and Apple long ago stopped supporting them.
If you have damaged the power switch and its connector, and possibly the connector on the logic board, there is only one way to repair. Buy a small momentary make push button switch, drill a hole in the case and mount it where it will not be obstructed when the case is refitted. Solder 2 strands of thin wire to the switch, leaving it long enough to solder to the logic power switch connector. As you refit the case coil up the excess cable. If you can't do this yourself someone with elementary electronics knowledge should be able to do this for you.