Contact the state of Michigan for informatin on orphanages.
They should be able to answer any questions. And go to their website--Michigan.Gov and search for the library. I find Michigans website hard and confusing to navigate but maybe you can figure it out. The library of MIch, holds millions and millions of records. There is an index of all they have on the internet. You can request copies of records, etc. and they will mail you anything you ask for free of charge.
Memphis Michigan is a pretty small town However there is a large institution that used to be a monistary for monks. and could have also been an orphanage at one time. The name of it is Sacret Heart and last I knew was being used for a drug rehab facility. It's very old and needs repair. It's out in the county kind of secluded. You drive dirt roads to get to it.
The biggest town East of it is Port Huron, Michigan. The largest town West of it is Lapeer. They had a large state mental institution in Lapeer. It's now torn down.
There was also a very large orphanage in Coldwater, Michigan. I believe it was called the Michigan Childrens Institute. Children were indentured from there. I am aware of this because that is where my grandmother and her siblings were taken when the state took them away from their mother. Then they ended up going to live with different families all over the state of Michigan.