Good that you are saving that stuff from now as I go for my ILR next month and I am struggling to scrounge up old mail. The thing is they want 10 pieces of mail, but no more than two from the same source which is problematic as I have reams of bank statements and not a whole lot of anything else. You don't need things with both names on (though I think it "looks" good to them), but you must have mail from both of you with of your names on it. BT must have changed their policy as my name was added to the bill about a year ago.
Save anything "official" dated with both years, as you need to show you lived together for the two years.
Utility bills
NHS letters
National Insurance Number mail
DVLA correspondence (I am hoping my license itself will be admissable as it has my name and address on it)
Bank statements
Credit card statements (if you only have US credit cards, change your address on at least one of them to your UK address)
Couldn't your Father in lawmake up a rental agreement?. I don't know how official they need to be, but I imagine a formally typed up document stating the address of the flat and both your names, rental amount etc would constitute an "agreement". Just a guess, as I don't know the first thing about property or any other kind of law.