Go down to your local council and ask to speak to someone in the Housing Benefit section - take with you your most recent bank statements (to show you aren't over the threshold for assistance, which is I think £6,000 in accounts... check though) and your entitlement to JSA.
They should fill out the form with you in person and it'll then go through the system and you'll get a letter stating what you're being awarded. This money will usually go into your nominated account every two weeks.. but bear in mind it's paid on a weekly basis (not monthly) and so although it works out at the same amount of money it can leave you short if you pay rent on a monthly basis - you could try and get them to pay the money direct to your landlord if this is easier though.
Hope you get the flat!
Edited to add: Don't forget to get your entitlement to help with Council Tax too