Any decent charter broker should be able to get you a small jet on an AOC to do this (do make sure it's got an AOC). If you really want speed then outbound from London City is super easy (though city reduces the choice of operators slightly). I have no connection with them but vaguely know some good people at Centreline or Xclusive jet but as I say any broker should be able to sort this.
At a guess under £10K from London City to Frankfurt, in a Citation, flight time; about an hour and a bit.