Two things:
1) I have no idea what you mean when you say "I have been looking for a really cheap sub at thrift stores or online. All of these subs have their wires spliced. "
2) If, as it appears, you control the volume from the volume control on one of your Bose speakers, you will have no way to control the volume of the sud from that. It will have to be adjusted separately from the Bose speakers. ...and that will get really old really fast.
that being said, you could get a sub with right and left line level inputs and use a pair of
"Y" connectors " on the back of your Bose speakers and feed one leg to the line level inputs on the subwoofer. But be aware of what I mentioned in # 2 above. But, if you constantly control your volume from the computer that should not be an issue.