Renting from an ex-partner is a very complex issue. It is possible but there are few hoops to jump. These hoops are designed to show to the local authority that you are now living in a separate "households".
I would suggest that you call
Community Legal Advice on 0845 345 4 345
They would be able to advise you directly on your entitlement and the application process. At the same time, call your council and ask for HB application form or check whether your council has one to download.
Sadly- nothing is going to happen by Wednesday; you can however apply to Social Fund (if you are in receipt of IS) to ask for a Crisis Loan {- ask CLA adviser about this too. You can download application here:
Apply for a Crisis Loan (form SF401) : Directgov - Do it online
Crisis Loan may help you in a tight spot but it may be difficult to prove that you are going to be kicked out. You may want to think carefully- would situation warrant a domestic violence angle (reembering that dv is not only about actual physical violence but emotional harassment and abuse too).
I don't know- this needs proper advice session.
With regards to your housing situation; if your landlady lives at the property and shares communal arreas with you (kitchen, bathroom, hallway)- you are not a tenant. Your rights are somewhat limited. You are a licensee.
CLA adviser needs to know this too.
You will probably need to make a homelessness application too- if your landlady is serious about getting you out.