Can I get sole custody without visitation?

I've had residential custody of my child since 2003. Recently, my child's father admitted to molesting a 13 year old boy and is currently serving a 6 month sentence for it and will have to register as a sex offender. He hasn't seen my child in over 2 years and has not paid child support in over 5 or more years. I've been trying to take him to court for the last 2 years and since I have not had an address for him, my attempts have failed. I now know what jail he is in and since he can't reschedule or just not show up, is it right for me to file a petition now? I've had a really hard time getting him to sign papers in the past, help with medical bills, etc., and most of all, I'm kind of worried that since the crime wasn't committed towards my child, custody will remain as myself having residential, or even joint custody. My child had stated that if she did have to see him, that she would like someone to be there with her, but at the time she does not want to see him for what he has done. What are my chances of getting sole custody without visitation?