The 19th and early 20th definition of “insane” differs from how the word is used today. Great-grandma’s admission to the “insane asylum,” may have meant that family members were simply unable to care for her at home any longer for one reason or another. Having her admitted to the county or state home or hospital may have been the family’s only option given her situation and theirs.
Records of these committals are local records–usually county court records, but that can vary from one location to another. There may have been a separate court to hear these cases, sometimes called an “insanity court,” or they may be filed with other court records. The person being committed/admitted is typically the defendant in these cases. The records of these committals, if they are open records, can tell you quite a bit about your relative at this point in their life.
If you are unfamiliar with how to access these records in the location where you person lived, contact someone familiar with the area’s records and ask them “if someone had to be ‘institutionalized’ in 1910 where would those records be?”