You're trying to feed the cable through the wrong way.

Most universal brake cables come with two ends. One is a short, round cylinder stuck sideways on the cable. That's a mountain end, and not what you need. The other is kinda bullet shaped, that's the one you want. Snip off the mountain end and feed the cable through the rotating bit on your aero lever, from the front. Carefully feed the end of the cable through the hole in the back of the lever (you may need a flashlight to see it) and it should pop out of the hole on the back of your lever. The bullet shaped end will seat into the larger hole on the rotating bit in your lever. You will have to pull the cable out of the housing to do this.

If you don't have the tools, I would really recommend letting a LBS do it. If they're not busy, see if they'll let you watch. It's very easy for the cable to fray into an unusable mess when cut improperly.