The licence fee funds the BBC
I'm sorry, but you're wrong: ITV does not receive any part of the licence fee. It should not: it is an entirely commercial organisation.
I am trying to find a documentary source for you, but I'm struggling. The law (Communications Act 2003 section 365) requires the BBC to collect the licence fee, but to pay all money collected (less any refunds to be paid) into the government's main bank account, known as the Consolidated Fund. The government then decide how to allocate whatever they receive.
The Consolidated Fund accounts for 2012-13 shows "BBC Licence Fee Revenue" as £3,122m. The BBC's Annual Report shows £3,091.7m income plus £16.8m 'premium' from the quarterly payment scheme.
The government, from 2008 to 2012, did top-slice the licence fee to fund Digital UK and the Switchover Help Scheme. Since the BBC was a large shareholder in Digital UK, its accounts were consolidated in the BBC's accounts, and SHS was also arranged under the BBC. Now that switchover is complete, that money is heading to the government's Broadband Delivery UK scheme.