It sounds as though she has been 'offered' the job, 'subject to a favourable reference from you.'
She has not really been dishonest with you. Not many employees would tell their employer that they are looking for another job. So I wouldn't hold that against her.
No matter what, she is going to leave you, so it would be better for everyone concerned to make things as smooth as possible. Write her a favourable reference.
You will have to pay her any holiday entitlement she is due anyway. So unless you need her really badly next week, you might as well let her take them as her notice period. There is nothing illegal in doing this, but I would advise getting her to sign something to say she requested it and you agreed.