How do I prove to the judge that my daughter's father is not capable of caring for our daughters?

Yes, the tapes of your ex saying he'd kill himself and could not care for the girls is admissible. You need to put the recording on a CD and bring a machine to court that can play them for the judge. You do this during your testimony. You tell the judge when the call message came in; you identify your ex's voice for him and then explain how his condition impacts his ability to care for them. Also make the point that if he cannot be honest with the judge about his condition (and his sworn declaration seems to be perjury), he can't be trusted with the safety and welfare of the minor children. I would like to see more in evidence than just the tapes though. Is there any other person who could come and testify to seeing him do (or not do) something that seriously neglected the girls' health or welfare? Like threatening violence in their presence? Or leaving them stranded somewhere that endangered them? Or leaving meds around where they can get at them?