Hey -
sorry to hear about that - such a pain when technology gets the better of the application process
Which KCL program have you applied for?
I'm a current UG student so have submitted a grad application to their GPEP 4 year course to which I have recently received an interview invitation. The reason for the background is that I wasn't asked for any additional references as I got a member of my current university (a lecturer) to do my UCAS reference therefore it was an academic one.
Is the reference on your UCAS form an academic one or a personal one? Most med courses say that if you are either currently in academia or graduated within the past x years or so (usually around 2) then your reference on UCAS should be an academic one.
The reason no-one is answering the phone (and I have personal experience of this as I tried to get in contact with them through the portal and on the phone all of Friday) is that they're moving buildings and having issues with the phone line. I would have thought the best thing to call the switchboard and be put through to the general admission department so that you can have an actual conversation with someone.
I have seen on various threads that some grads (but not others) were asked to upload their UKCAT scores to the portal so KCL are obviously having some technical/organisational difficulties. Hopefully this would mean they are more understanding to your situation.
A backup option is to send an email to kings via the portal with your reference attached to it then at least they have it either way.
I'm not sure if any of that has been much help but thought I'd pass on any information I had as your situation isn't really your fault!!!
Good luck with it all