It is far better if you can get the airport staff to deal with rebooking your tickets than trying to call TK CS because most likely TK CS will have no authority to rebook you to other airlines. Airport staff can do that. I suggest that you keep trying to get hold of staff at the airport to help you. You can also email the local TK ticket office for assistance; they should be aware of the cancelled flight, rebooking options etc.

Anyway, if you end up staying 36 hours before you are able to get out of there, you should keep hotel, meal receipts and ground transport for later reimbursement from TK or your trave insurance.