MIL has Alzheimer's and is adamant that she will decide when to go into assisted living.

MIL wants to decide. Wants to have control over her life. (Who doesn't?)

But Alzheimer's Disease has come.
Has taken much & will continue to do so. Alz will decide.

Discuss MIL's care level, insight & ability to have capacity to decide her living situation with her Doctor. The POA will take over when required. The POA will then make arrangements as needed.

Sadly, you can't reason with people who have losing/lost their reason.

When I toured facilities I asked each place when they felt it was the best time for a person to move in.

Each place told me that it was best for a person to move in before it became absolutely necessary to do so.

They were able to adjust better to their new surroundings.

You ask "when do we say it's time," well..... like now. Your MIL no longer has the ability to make good choices for herself so you must step up and do what is best for her and her safety.
Just make sure that whatever assisted living facility she moves into also has a memory care facility attached as she will be there eventually, perhaps sooner than later, and you'd hate to have to move her again as that is very hard for someone with mental decline.
You now have to tell her how things are going to be as you will now be in the "parent" role.
Best wishes in finding the best facility for her.

Sounds like a lot is going very well.
POA ✅️
MiL has home care already ✅️

But the next stage is looming.. you are wise to plan ahead.

IMHO temperament plays a part.

Some folk will simply trust their son or daughter when they say 'it's time to downsize' or 'move to a new place where there is more help'.

Others will say their kids are plotting, stealing, taking their house. (Paranoia can happen, or they are very independantly made, or have lost more insight)

Sometimes a trusted Doctor can say "it's time" with good success.

I've had friends go through these sorts of scenarios above.

Having the downsize/more care chat - I was advised that it takes At Least 6 times to have this kind of conversation before a senior will hear you. For it to sink it. To even start thinking about change. That is WITHOUT a dx of dementia.

If there is cognitive impairment to cause lack of insight into their loss of skill/function level & therfore their care needs - how can they understand?
That's the level my LO is at. Dr called it 'Anosgnosia'. It looks like denial but differs. The brain really just can't 'see' the issues.

So where are you at Momsthing?

Deciding HOW & WHEN to have the chat?
Hoping for MIL to'see' & decide herself or give permission?
Deciding WHO actually has the authority to make the move?
Or decided the time is now - need the HOW to make it happen?

See her neuro-psyc and get both the letters needed and his/her agreement that mom is no longer safe at home. Then have the honest sit- down talk. I am afraid it will have to be as brutally honest as it needs to be. Don't expect her acceptance. This is a matter of need rather than "want".

I am assuming she doesn't live with you and is alone?.
I think you already assume she is unsafe?

Time to do what's best for all involved. She has a broken brain and can no longer make good decisions.

There is no easy way to handle this, when we had to move our step-mother into MC, she said "I don't want to be with all the crazy people"!

Well, we stood firm and she is in a very nice MC, and well taken care of. My brother & I have her POA!

Can you get her doctor to tell MIL she needs to go to a facility and also have the doctor recommend one to her ( that you have previously chosen) ?

She has already decided, she's not going into one.

You said MIL (mother in law ) doesn’t think anything is wrong with her . Very common with dementia . She will never agree to go to a facility . You have some options . You tell her you are having lunch and tour a facility when in reality you are bringing her there to live .

Or , Find a reason to bring her to ER , and tell them she can not live alone and have the social worker assist you in placing her in a facility . Tour some ahead of time (with or without mil ) and have some in mind .

Or, You call your local agency of aging and they will send a social worker to her home do a needs assessment and if she is deemed not safe alone and needs 24/7 supervision , they can remove her from the home to assisted living. Have one picked out . MIL is not going to come around and go willingly . Good luck .

"It's almost like she doesn't believe there is anything wrong with her except "old age", she says".

Picture breaking your foot.
You see it all swollen & you FEEL the pain! You can reason it is sensible not to walk on it. You can reason you will need help.. crutches, a wheelchair even.

Picture small parts of a brain getting damaged by disease.

Alzheimer's Disease impairs memory, thinking and behaviour.
It disrupts the brain’s neurons, decreasing messages travelling normally through the brain.

You can't see it with your eyes.
People can look the same for quite a while (although it does age them as it goes on). This makes it hard to remember they really CAN'T think as they did.

I have family grapplIng with this.
Respectful, offering choice? VS possible denial in the caregivers?
It's a fine line.