Suggestions from a distant LL:
How frequently does she now pay rent? Monthly? Weekly? Matters a lot as to required notice.
A LL is supposed to follow due process before a tenant gets tossed --read up as to AZ!
Yes there is a lease--an oral lease! At least until some court hold otherwise, which I doubt will happen given that she paid and he accepted rent. .
Now if somebody told me I must sign a lease I'd be very polite but keep my yap shut except to be nice and look at lease and tell them I need to review the terms and I'll get back to them. I'd not be nasty and I would NOT comment to LL as to my view of facts or law.
If LL hands me a lease but no notice he has already dropped one ball--but do NOT educate him.
Were I the LL for sure I'd want a written agreement to protect me--but that doesn't mean I'll get it!