Does the bible explain anything about the mentally ill?

It doesn't mention mental illnesses by name but psychology didn't exist when it was written. It does say that sin is in our nature, so it doesn't oppose the idea of genetic predispositions to behavioral problems. However, as for the woman you're talking about, not everyone who quotes scripture is saved. Satan quoted scripture to Jesus while he was in the wilderness. If she is mentally ill, don't hate her for it but you don't have accept any of it is truth.