The fact that he lied about owning property is gross, but if he had told her, why wouldn’t she contribute to the monthly bills? She is occupying space.
Again not telling her is shady but if she could make an informed decision, paying rent to live in a house isn’t crazy, even if one person is accruing long term value from the spend.
If it were me I’d obv tell her day 1, then offer a generous rent rate. The house is being worn down by 2, but you are gaining long term value (paying principle on the mortgage). She can’t expect to live rent free, but you can’t expect her to share the mortgage burden equally.