On Monday, if not today, you go to the school and make an appointment with the school counselor and principal. You show the note and you fill them in on all the ugly details. You tell about the violence in the home, the drugs, the mental illness, the child hiding under the covers. You give them copies of the note.
They are mandated reporters, they will report it to CPS.
Then you leave Lexy at school (and they will probably have a nurse evaluate her and talk to her), and you go to CPS and you talk to a CPS person and tell everything.
Stop trying to get your daughter committed. The solution doesn't lie there. Her problems are so deep she'll have to be committed several times and cycle in and out through the system and try this drug and that and this treatment and that and this program and that. And none of it will stick because she doesn't even want it. Lexy doesn't have time for her mother to get better and learn to grow up and parent responsibly.
Don't go to the school and protect your daughter or her BF or try to figure out how to save everyone or keep everyone employed or keep their apartment for them. Just focus on Lexy. And also stop worrying about the injustice of your daughter blaming you for her problems. Blah, blah, blah, what else is new?
Focus on Lexy and having her monitored by the school and by the state. Make it clear to CPS that you are available. When you go to CPS ask for services not for your daughter but for YOURSELF, ask for what's available to you so you can be a support to your granddaughter. Let them know that you realize you have a massively dysfunctional family and you want to do YOUR part in fixing you, so if the time comes and Lexy has to be placed out of her home temporarily or permanently, you will be already approved by the state to take her in.
You might not be able to afford a lawyer, but the state can appoint one for Lexy (a guardian ad litem), and that's pretty much the same thing. Ask for a GAL for Lexy. Ask for services for Lexy. Ask for monitoring. Ask for evals. Ask for family counseling, family services, family support.
If you talk to the school the school will be on alert to watch for signs of neglect and stress and violence. They'll notice if she's clean, fed, absent, and picked up on time. Frankly schools can't do a lot, but they can witness for CPS and they are considered an unbiased witness.
You worry about your daughter losing her job and apartment. So what if she does? It just puts more pressure on her to leave Lexy with you while she finds another job, another place to live. It just makes it clearer to CPS and ultimately a judge that she's unstable and unfit for mothering in her present condition. As long as you enable your daughter and her boyfriend you enable them to mistreat Lexy.
By not protecting them from the consequences of the ugly truth of their lives, you are protecting their ability to neglect and mistreat Lexy. The way to protect Lexy is not to protect them.