Could I just say to 11minus that I'm very sorry to hear this about a fifth rejection but also .... if your DD is in a prep as I think you say, what on earth were they thinking advising you to have Harrodian as your 'back up' option. Fgs. Everyone knows that school, as one head once phrased it to me, is "quite frankly, away with the fairies." Another told me he had "no comment" about Harrodian because it's as if they pick random names out of a hat.
They could have suggested somewhere like More House, or Radnor House, Marymount, Claremont Fan, St Catherine's, Twickenham, St Johns, The Hall Wimbledon, Notre Dame - any of those.
As others have said, you can try maybe Fulham School. I know Kew House opened a new school recently in Maida Vale? Also, maybe try Radnor as they sometimes have places at the end of the process and end up phoning around. Your head needs to get on the phones asap. I'm sorry you are in this situation.