3.5mm audio jacks include 1 or 2 signal lines and a ground. Hopefully there isn't any looping happening on the sound card or amplifier and speaker assembly.
I think you're hearing the magnetostriction effects from either the 120 Hz rectification or the ~kHz switching, from one or all of the PC power supply, the monitor power supply, or by way of induction from the environment, meaning 12VAC cords, nearby equipment or wiring in the walls, to the preamp section (sound card, cord, etc.). This can be verified (or contested!) by measuring the frequency of the hum with the meter's frequency counter feature.
Try the monitor with another computer; try it with an outlet on another circuit (ie: supplied through a different breaker) -- probably not the same wall. Check and play with the power supply connections (12V, 5V) to your motherboard, and, if they're separate cards, the sound and video card supply connections. Wiggle the PCI[X, e] connection, too, if the sound or video card are separate. This is just looking for poor power contacts. Minimize the distance the video line (PC to monitor cord) lies parallel to nearby power cords. Use a digital video interface standard like DVI (in digital mode, since it can carry analog VGA, too), DisplayPort or HDMI instead of VGA (analog) for error protection and correction.