I returned home from holiday earlier this year to a warning via letter that my tele bill was abnormally high. My son denied all knowledge of how this could have happened.
Long tale, but this was achieved (?) by my son (who should not have been in the house), surfing the net, viewing a porn site - agreeing to terms and conditions. Software was then automatically downloaded onto pc, the free internet connection disconnected and a new connection via the premium rate made.
He made 2 or 3 connections and it cost me over £200! I was hopping mad. He felt you have to have use a credit card to run up bills for porn. Not so!
BT were able to give me full details of the number dialled, including the fact it was 'Night time media' head office in New York. Not much chance in contacting them to complain. My son was not even 18.
Premium rate numbers are banned in my house. I am sure premium numbers can now not be dialled via the internet or am I mistaken? I would be surprised if they can still be dialled through your isp also?
This may not be how your high call cost arrive, but something to consider. I am surprised BT are unable to tell you who runs the number. I am sure it was a special department via BT who gave me the information. I would try again.
How old are your children? It might be worth having a word with them to see if they can shed any light on the matter. The premium rate calls could be via the landline or Internet. I would not cease the internet - find out how it happened, as I did. I feel very secure in the knowledge it is highly unlikly to happened again (hopefully). My son would not dare after the trouble it got him into!!!!
Good luck and don't panic.