How do you go about finding an age care facility for my mother near her home in Sunshine West?

I used APlaceforMom.com with excellent results. They gave me multiple recommendations of places to look at, and from those I found a perfect fit for my mother.

Your Area Agency on Aging can help you locate a suitable facility and will also send out a social worker to assess your mom's needs. Call them.

You'll probably need to arrange an ACAT assessment for eligibility, but check the MyAgedCare website...

https://www.myagedcare.gov.au/find-a-provider/

Enter a postcode, type of service required, and facilities nearest will be readily available.
It's more important to find the most suitable facility, but also being close to those who will visit would be preferable for convenience. Be sure to tour the facilities before reaching a decision and do read about all the fees and conditions prior to signing anything.