What you have is a fairly concise story. Back before WWII having people surrender their children to an orphanage or to a relative was common. It could have resulted fromthe death or incapacitation of a parent or from pauperism which was widespread.
You are cut off from tracking these girls in the 1890 census because it was destroyed in a fire. The records from this asylum could exist but trying to figure out where is a problem. And what the records would contain might be little more than you already have. The girls were in that place less than a year. If you want to know where it was in Cleveland it should show up in the 1900 census.