Yeah, this is a tricky problem unless you want to pay $40 for an adapter from Microsoft.
There is a pretty simple hack to get it to work, though. You can't plug the standard AV cable and the HDMI cable in at the same time, the way that they are designed, and since a DVI connection only carries video, not audio, a simple HDMI to DVI cable won't do the trick either. The easiest way is to take your original AV cable (the bulky gray one) and use a flat head screwdriver to pry the plastic casing off. Don't worry, this is safe, but be careful not to pull loose any wires. Once the casing is gone, you can plug the HDMI to DVI cable from your 360 to your monitor, and then connect your audio via the standard AV cable, now that they can both fit!