I am legally unqualified: If you need to rely on advice check it with a suitable authority - eg a solicitor specialising in landlord/tenant law...
Sounds like you have officially let your house to this lady whilst she pockets the housing benefit and stays at her mums house. If you have both signed the AST then your up shits creek, she is unlikely to sign any end of tenancy form because she then stops getting paid.
Report her to the council for benefit fraud, and issue a section 21 to regain possession.
IF you can get her to sign a form giving you permission to discuss their claim with the council then you have a chance to get paid direct by proving she is in arrears (2 months minimum i believe) but i think this is up to the council discretion whether they will or not. again she is unlikely to agree to signing such a form.
If she simply doesnt want to move in then get your keys back (and change locks to make sure she hasnt made copies), then get her to sign an end of tenancy form which will allow you to rent the house again
The opinions I give are simply my opinions and interpretations of what I have learnt, in numerous years as a property professional, I would not rely upon them without consulting with a paid advisor and providing them with all the salient facts
If she wont sign any forms for end of tenancy then the landlord has to issue a section 21, you cant just re let the place when there is a tenancy agreement in place surely? if so then i stand corrected.
i would view it the same as abandonment, she could have moved in for one day for all the landlord knows, trying to prove she never moved in at all (especially after issuing keys) would be mighty difficult.