What are the steps to switch my daughter's student visa from a language school to an international school in Thailand?

well seeing as she's on an extension that's only good until June 14th and the new school isn't going to give paperwork until June 16th, you either get another 90 day extension with the place she's enrolled at now or you run on overstay for the time until you can apply for a new extension with documents from the new school. <- IF your immigration office will let her overstay until she can apply for the new extension (you'd have to go talk to them before you try to do it that way)

Your daughter isn't on ANY visa at all, and the stamp right now that lets her stay in thailand says "extension of stay permitted until" and has a date (which you say is June 14th)

I'd say get an extension with the documents from the school she's attending now, then get documents from that school stating she stops studying on xxx date, and get the new documents from the new school where she will be attending.

You go and cancel the current extension and immediately apply for a new extension with documents from the new school. All you're doing is changing from an extension based on education at a private language school to one based on education at an international school.

Shouldn't have any problem doing that "reason change" on extensions