The issue is that you need something to decode the audio signal from a single stream into 5.1 channels. It's not just a digital to analog conversion that you need.
There are decoders available. I just did a quick Google search and found one for about $130. There might be less expensive ones. If you were going to spend that kind of money, I'd go for a surround sound receiver (like maybe the Onkyo 309 for about $180). You could connect your speakers to that receiver, and connect the subwoofer to the subwoofer output on the receiver. You would need to convert the RCA connection on the Logitech speakers to one that works with the receiver, but that's not hard (you can buy them premade).
Now we're talking about buying new gear, though. Is that something that you are interested in doing?