Well, tried this today, on both galv. and SS cables, and so far, no joy. Solder just balled up. Using Kester 60/40, Kester paste flux and liquid flux.
Huh...no idea what might be the issue if you followed the stuff above. The cables in the pic are, in fact, SS (Jagwire). Maybe yours are some sort of pre-preg with silicon lube or something? Really douse it with alcohol and then let dry completely before dipping in the flux (alc dilutes flux immediately on contact, so if there's a film of wet Alc on the wire it will not allow the flux to penetrate and do its' job).
When was 60/40 leaded solder outlawed? Didn't know it was...news to me.
Kester Solder | Mouser