What is the ruling on taking one's wife's mother to a mosque that is narrated to have been built in one night, and is this shirk that invalidates the fast, and what is the expiation for it?

If going to a specific place for healing is considered to be from Allah Almighty, then this is an innovation (bid'ah) that does not reach the level of shirk (polytheism). Repentance is obligatory, and one must not return to it. This has no effect on fasting, and there is no evidence for a specific expiation for it.