Usually Windows hides system tray icons after a while. Try clicking the upwards-pointing arrow that says “Show hidden icons” when you hover your mouse over it. In the dialogue that appears, you can click “Customize” to choose whether icons are always visible.
Also in that dialogue, click the icon that has the green checkmark “Safely Remove Hardware”. That should list the Alesis if it is recognised by the computer. If so, click on the Alesis item then Windows will tell you when it is safe to remove the USB cable from the computer.
This is the correct way to remove a USB device. It is very important for something like a USB external hard drive. For a USB audio device it is usually OK just to disconnect the USB cable.
I would have thought Windows “Safely Remove Hardware” would have made the Multimix safe to remove, however this Forum is for Audacity, not Alesis. If the mixer needs an operating system shutdown in order to physically remove the mixer from the computer, hence they give you no system tray icon, you have to do what they say.