In order to use built-in speakers on a monitor, you have to have a cable connecting the monitor to the audio port (either the onboard sound chip or a sound card you installed). Depending on the monitor, it's either a separate 3.5mm connector or it might use an HDMI connector.
Ideally, what you'll want to do is have the monitor connect directly to the main audio output, & connect the headset to the headphone jack (front or top of the case).