Joint tenancies

Rent Arrears

We had two tenants on a standard AST joint 6 month tenancy.
On the last day of the 6 months, one of the tenants gave one month’s notice but failed to pay any rent to cover that period.
The remaining tenant continued to pay her half of the rent until she left 3 months later.
When she left the cost of the reparations to the flat exceeded the value of the deposit. No refund of the deposit had been made to the tenant who left first.
What is the legal position with the tenancy if only one tenant gives notice and the other remains. Is it then considered a separate lease?
Presumably the tenant who gave notice is still jointly responsible for the unpaid rent for his notice period.
When the first tenant left, should the agents have completed an inventory check (they completed one when the second tenant left).
Does he have any liability for the unpaid half of the rent until the remaining tenant left.
Thanks for any advice