I'm not familiar with your particular kit but I'll tell you how I do it with my TD20.
The TD20 has multiple audio outputs so I route these to my recorder (it's a stand alone hard disk system) and play live into it. I don't record MIDI messages. To do this on the computer you'd need some form of audio interface to plug the analog outputs of the drum module into. These range from inexpensive 2 channel USB units on up to expensive 16+ channel ones.
Your other option is to record the MIDI output stream into your software via a MIDI interface (some of the inexpensive USB audio interfaces have MIDI ports also). This would allow you to edit the MIDI data directly (to be honest, I don't know enough about Audacity to determine if it will record MIDI data vs. audio, but most commercial recording packages do). After editing, you then route the MIDI data back to the drum module and it plays the module, which you then run into your mixer.
Each method has its advantages/disadvantages.
Here is a tune I recorded a few years ago with the TD-20. I played all parts (drums, mallet percussion, bass). The Return of Wayne Dangre
The MP3 quality isn't so great but you can at least get an idea.