Why would you want to mix externally, and then only deal with a simple stereo signal into the computer? The best use of computer power, would be to have separate digitized audio tracks - so you can use the appropriate multitrack software like sonar, cubase or protools to process each track separately and then mix them on the computer.
You don't need an external mixer for this. You can either record one instrument at a time using the line in port of your sound card, or purchase either a multi-track internal sound card or mutlitrack analog-to-digital external hardware, like protools.