It sounds like your daughter has mental health issues that cause her to be severely irresponsible. It is surprising that at least she is not diligent about seeing her son. I was going to say to call the father until you told me about how irresponsible he is. Your daughter may have a level of immaturity that is hindering her ability to be a parent. This may not be actual choice but a lot of people lack of the skills to be good parents. It is going to be difficult if she isn't willing to do anything. There are parenting classes for people who lack these skills. Nothing is going to change unless she can connect with someone. She has to want to be a parent. You have to find something that will mature her. If not then you can petition the Court for grandparents visitation or rights. She may be threatened at the thought of someone evaluating her because she knows it will reflect on how others see her. Reintroduce your offers to make counseling more affordable until she sees that she needs help. Someone may change their mind very quickly when faced with such consequences. Investigate mobile mental health units in your state where they come to her.
It sounds harsh but this situation calls for harsh responses. This child is at risk. I know you feel guilty but dad isn't an option. The child will just be an inconvenience. Instead of calling dad you need to practice tough love and call Child Protective Services. You could make an anonymous call and they will remove the child. It is for his own good. You made a critical mistake however. Never threaten to do anything that you are not prepared to carry out. That gets across the harshness but it isn't going to happen and it shouldn't. CPS may call you or other relatives but at least he will be cared for. And that is her wake up call. I think your plan at the end makes a lot of sense. Stop enabling her and start with tough love. But be prepared to do what you say. If nothing changes call CPS.