It might be better for both you as the landlord and Ms Tenant to keep the periodic tenancy ongoing, giving you the option of giving her two months notice, or her giving you one months notice. She might not fancy staying there much longer, memories of now dead relationship etc.
Maybe she is about to apply for Housing Benefit, paid to her, to help cover the rent?
It sounds like Ms Tenant will now be the only adult, in which case she can make her money go a little further by notifying the local council of the new situation and getting a 25% Council Tax bill. Equally, she may be eligible for Council Tax Benefit.
Has Mr Tenant left immediately before rent day? Is he giving Ms Tenant his share of the deposit, to help cover the ongoing costs following his “sudden” leave? It’s certainly a complicated situation with one tenant leaving while the other appears at the moment to want to continue being there.