If you don't know which institution she was committed to, or exactly when, and are not sure who, or what organisation, committed her, this is likely to be a tricky project.

Firstly, you will need to bear in mind that the records you want are still relatively new, in archives terms, and it's possible they won't have been transferred, at least in every case, to a central archive and will still be kept by the institution concerned. Secondly, as you are probably already aware, mental health records are pretty sensitive documents and – at least in cases where the person concerned may still be living, or have close living relatives - there may be significant restrictions on who is allowed to access them, which may not extend as far as a great-nephew relationship. You might need to rope in other members of your family to make enquiries in some cases. At least, that's certainly the situation where I'm based, in the UK, and though the position in the US may be different, I'd expect that you will definitely find that there are privacy issues to confront.

You'd need to start by compiling a list of places where she might have been sent, and this will probably have to be done on a county-by-county and city-by-city basis. In addition, you'd need to search for any federal facilities, because these, if they exist, would have a completely different record keeping system and have records that woulds be stored in federal, not state, archives. The key term used by the local authorities & local archives seems to have been "Mental health custodial facilities", and hopefully that can be used to start unlocking a few doors when you start searching, so's to speak.

With that done you could make enquiries at the New York State Archives in Albany, which is a huge and sadly under-funded facility with a rather limited online catalogue. It stores vast quantities of records, many of which are in off-site warehouses and which can be very frustrating to obtain without knowing precisely what you're looking for and precisely what the file references are in advance. In my experience, which admittedly is a decade old now and which involved searches for century-old trial transcripts, not mental health records, the NYSA is also not massively responsive to vague email enquiries - hopefully that situation has improved since I dealt with them.

Luckily the NYSA does have an historical overview of mental health in New York State, 1945-1998, together with a "New York Mental Health Documentation Project" that is gathering and cataloguing records, and it has produced a short preliminary guide to these resources which you can access here. But having reviewed this briefly, I have to say it's quite a dispiriting document, not because there are no records but because there are so many and because they are on paper and microfilm, not digitised - meaning they would need to be searched on the ground and by hand. If you aren't sure where your great-aunt was treated or exactly when she was committed, this would be an enormous task.

If I was undertaking the project myself, I'd start by writing direct to the institutions on your master list to enquire about their records and conditions of access, and to find out if they can actually tell you if a person of your great aunt's name was ever one of their patients. But even this would require quite a significant effort. I quote from the NYSA guide:

In order to find and use potential records, the researcher will need to locate the appropriate department within the local government (such as the county or municipal department of mental health or local government sponsored hospital with a mental health unit) and discuss information needs with the department's records management officer.

As I say, without some way of narrowing down the number of potential contact points you may be dealing with, this alone would be a considerable task.

Good luck!