We were rethinking about the "sit down" but we are still going to do it. Her mother has gotten more verbally abusive. My brother and his girlfriend yell at my niece whenever she ask for something. Her mother has said that she can't handle my niece anymore. My niece has ADHD. My niece she a very hyper child but if you keep her busy, she does really well. My brother and grilfriend love to play video games, watch tv, and sleep. They basically never want to do anything with her. We all have seen this behavior.
If you're concerned about the child, forget custody for the time being.
Approach it in this manner: mom is older, not doing well with her health, and WE were wondering how we could help mom, you guys, and little LuLu (just a name to represent the girl, don't want to use her real name) more.
Mike (a name for your husband, protecting his identity, too) and I were wondering if we could watch sweet, little LuLu for 4-5 days a week, more if mom needs us to spell her?
Mike and I were thinking we could all try this for 30 days, and talk about it after 30 days and see if mom needs more rest, you guys, too.
Don't mention custody, because if they approve, the child will be properly patented, schooled, fed, loved, and get medical care.
Then you go slow and nice, get it last at least 6 months and then see what happens.
As Proserpina said, the last thing you want to do is get pushy and make enemies of them.