No, it isn't unusual and you aren't wrong. To be 'on the mend' is a perfectly normal everyday expression, and it isn't specifically British. If you search this expression online, you will find definitions in American dictionaries and US language sites.
I've no idea why your teacher is unfamiliar with this expression. If she is quite young, she may not use it herself, but she should at least be aware of its meaning.
Are you absolutely sure that this woman is who she says she is? I'm afraid that it is not uncommon for non-native speakers to claim that they are native speakers, when in fact English is their second language. These people will often use the excuse 'Oh, that's British, and we don't say that in the US' to cover up the gaps in their knowledge of English.
Even if she isn't actually lying about being a native speaker, her English may not be perfect. For example, she may have been born in New York and lived there all her life. But if she has grown up in a Latino neighbourhood of the city where Spanish is spoken all the time, she may not have a fully idiomatic command of English.
These are just a couple of suggestions. Without knowing more about this person and her circumstances, nobody can give you a more precise answer. But be assured that there was nothing at all wrong with the expression you used. That is exactly what many native English speakers would have said in that situation.