Sorry to hear about your job. This is not an easy situation to explain to the benefits people.
From what you say, you live with a partner. His income may affect your benefit claim. If you've been working for two years you should be able to claim contribution-based JSA irrespective of partner's income.
To claim help with rent or council tax you would have to demonstrate that you had a tenancy, so a tenancy agreement and rent book would help. It would also help if your name was on the council tax bill. However, unless you had your own room your partner's income would be taken into account.
If your partner is on a low income you may be able to claim Working Tax Credits but this is dependent on hours worked and income. There's a calculator at
EntitledTo.