My mother is mentally ill and making terrible health choices, what power do I have to admit her to a hospital against her wishes?

If you don't have PoA for mom, all you can really do is play the waiting game.

She's going to fall and wind up in the hospital (chances are)...and then she'll probably want you there...IDK how your relationship is with her.

If/when she winds up in the ED you can report her as an unsafe discharge and hopefully get someone to check on her living conditions, etc. She may 'lose' the right to choose for herself.

In the meantime, let her know you're there for her, but sounds like she just does what she wants.

Curious-how old is she?

No, you have no power to get your Mom to do anything from hospital to home care to rehab. You can only suggest it. We have a right to make our own decisions until we are adjudged by medical and courts incompetent to make our own choices. That will NEVER happen due to substance abuse, and is unlike to occur due to mental issues.
Recommending a book, a memoir by Liz Scheier called Never Simple, about her lifelong journey attempting to get care for her mother who had mental issues and substance abuse. Ms Scheier and the entire social services community of the city and state of New York wasted their efforts until her Mom finally died. The book is a wakeup call on what can and cannot be done. Consider Al-Anon where you will get great guidance, ideas, support and information.

If you (or no one) are her PoA, then you will either need to pursue guardianship through the courts and prove she is unable (not unwilling) to care for herself, or you keep reporting her to APS until the county acquires guardianship of her if she is unable (not unwilling) to care for herself. Even if she has a medical emergency and goes to the ER, you can tell the staff she is an "unsafe discharge" but if she seems to have capacity, they will let her go.

I agree you need to go to Al-Anon for yourself to find reality and boundaries. Yes, alcohol is a symptom but it is the main thing in preventing her from dealing with her deeper issues. If she's also mentally ill, she won't be able to take any meds safely and effectively until she's off the booze. So, it is the first thing that needs to be dealt with.

You seem to keep wanting your Mom to be someone she is not, and probably never was. Please know that you aren't responsible for her happiness. You can't have her recovery for her. These are hard truths, so I wish you much clarity and wisdom, and peace in your heart whatever the outcome.

If she has a doctor, let him or her know about mom’s condition and situation. HIPPA laws prevent the doctor from talking to you, but you can still write an email or letter stating your concerns. Then make an appointment for mom to have an updated medical evaluation, lying to her if necessary to get her there. See if she will agree to preparing documents such as POA and an advanced directive. If none of this pays off in getting any help or progress, not your fault, know that you’ve joined a common club here, waiting on an event that will force change. The event will come, it’s no fun waiting for it, and providing an illusion of independence for her will only delay the event. I’m sorry you’re in this position and wish you the best in coping with it

Start by going to Al-Anon.

You are waiting for a fall, as mentioned above.

Sorry your mom is such a mess! Do you have POA for your mom? If she falls, call 911 to get her up and get her evaluated. She's a train wreck looking to happen so I can't imagine it will be long before there is an emergency. Who takes care of her dog if she can't? Is anyone caring for her or is she just muddling through on her own?

So at this stage your Mother has refused a 30-day program & also local hopsital treatment for her migraines.
She has not yet hit her rock bottom. Not ready to make any changes.

Indeed you are awaiting the next crises.

There is much here you cannot control. Work on what is within your control.
Be ready for that next crises. Be ready to inform her care team of her recent choices. It's up to Mother to change her mind.

You need to admit your mother is an addict to cigarettes and alcohol. Her depression and migraines will not improve until she seeks treatment for her addictions. You need to go to Al-Anon to better understand her addictive behavior.

A hospital is probably not going to offer or order home assistance for her.

What do you hope will be the outcome if Mom is admitted to hospital for treatment?

I am 66 and I don’t walk on ice, …. You mom doesn’t belong walking on ice at age 70, regardless of her health..