Welcome to the forums!
If it wasn't clear, the notebook will automatically switch between speakers and headphones depending if anything is plugged into the headphone jack. There are no settings or way for you to control this.
So....if you get sound through your headphones but not through your speakers (the notebook is no longer automatically switching the audio output for you), apparently there is some hardware problem related to your notebook speakers (which may or may not be related to the speakers themselves).
As dstrakele (thanks for all dstrakele's help!) mentioned above, there is a chance that the problem is with your headphone jack that is keeping the audio 'stuck' on headphones that you might be able fix.
Otherwise, if you have any interest, you can also send it to ASUS for out-of-warranty service although you might also want to double-check if you have any warranty left. Let us know if you want more information about this.
If you don't mind connecting external speakers, this is another idea.
Good luck!