"Tenant" without AST

Possession - Section 8 and Section 21

I allowed a lady to live at a property I am selling to her - we agreed a "Holiday let" all inclusive for one month, which is the time she wanted to arrange her finances.
THis did not happen, but she was keen to put the utilities into her name, which she did. We agreed a basic rent, again she stated this would be only for one month. three months down the line she is still no nearer to either paying a deposit on the house, or finalising the purchase.
Question: I want her either to exchange or get out. We do not have an AST. Can I (or should I) serve section 21?
Do I need a belated AST?
She gave me £1000 as security deposit. THis has not been registered - any advice?
Thank you.