My brother is 12 years old and I'm 19, we both live with our mom, but she is mentally ill and refuses to get the help she needs. She is depressed and drinks and goes crazy to the point that cops are called, she is hospitalized, she tries killing herself, all in front of my brother. The intensive outpatient she is in right now isn't working at all, so if she refuses to get the help she really needs, do I have a good chance of getting custody of him? He has already saif if given the choice he would choose to stay with me. And I have a full time job with benefits so I could afford to take care of him.

Is the boys father in the picture at all? He may be first in line for custody before your considered.

First you are still living with mom, so you at this point and time offer no better living conditions. So you will need to move out and set up your own home, living on your own and paying all of your own bills.

Next could or would the mother be willing to sign guardianship over to you without a court battle ? If that is yes, it is easier, if not, you are looking at 5000 to 10,000 just to start in legal fees.

The other choice is to call social services and get them involved, they may take the child out of the home, and if you have your own home, then you can petition them to let him stay with you, while your mom gets the help she neeeds.

If she was in the hospital, did the doctors put down in writing that she had a mental problem, and you will of course need written proof of everything you are saying,

And as Califdadof3 said, the bio father will have the first and best right to ask for custody of the child

You could approach any social worker type person on her case. But as stated, the father has to be in the picture if there is one. Keep in mind that having a job doesn't mean that you have shown the family court (that would approve guardianship) that you can afford to live on your own with the added financial and time burden of caring for a brother. You need quite a bit of savings just to get an apartment and furnishings, and that's just for starters.
Your mother, meanwhile, might qualify for some Medicare home help, someone to assist so that you don't carry the full burden. Usually this coverage is for personal care like bathing and so on, but look into it for cooking and laundry and housework. Many locales have a non profit agency that fills in the gaps for what Medicare doesn't cover.
It's possible that your mother could benefit from rehab, a totally different environment from inpatient or outpatient hospital programs for the mentally ill. It's just that most of them cost rather than being covered by Medicare. Of course she may not be let in if her mental illness is too severe. She is being allowed home at night to care for a 12 year old, which lends me to believe that she could get into rehab.