The Dowler family was told shortly before the trial of her killer earlier this year that journalists had hacked into her phone and deleted messages.
Chief Constable Mark Rowley said he could not comment while the Met Police phone-hacking investigation continues.
He said he did not want to undermine the investigation.
On Wednesday, Milly's parents Bob and Sally, along with her sister Gemma, met Prime Minister David Cameron in Downing Street over the phone hacking claims.
They have also had meetings with Labour leader Ed Miliband and Deputy Prime Minister Nick Clegg.
Speaking to BBC Surrey, Mr Rowley said: "I am not going to get into detail of who knew what, when.
"I am immensely frustrated about my inability to put all the facts out there.
"But I'm sure you'll understand I'm going to be very determined not to undermine the Metropolitan Police investigation."
Levi Bellfield was given a whole life tariff last month for murdering 13-year-old Milly, who was abducted from Walton-on-Thames in 2002.
Following the trial at the Old Bailey, Mr Rowley criticised the way the Dowlers were cross-examined.
He said he was "shocked" by their treatment and called for changes.
Surrey's police chief has said he is "frustrated" he cannot respond to claims his force was aware that Milly Dowler's phone had been hacked.