I would question whether or not the fan was really at fault. The charge of £665 seems like a lot for a simple fan replacement. We had lights rewired and installed for far less and our electrician wasn’t the cheapest (but came highly recommended) either.
The landlord agreed to a quote and that’s it. Based purely on the OP’s account there was no explicit agreement for repair work (only initial assessment) to be carried out. The landlord should not be liable for assumptions made by others. Maybe a split cost at best is in order.
A landlord isn’t there to be your doormat. Why should they pay for work that was never sanctioned by them? I’d be furious if my tenant tried to charge me for work I didn’t approve unless it was a genuine emergency (which this wasn’t). From what I read a single circuit was blown. Assuming the circuit breaker is of the magnetic type (sounds like it) that would indicate a wiring system that isn’t too ancient. This means the key kitchen electrical appliances (stove, fridge, washer/dryer, etc.) should be on separate circuits to the common power sockets. Therefore, where is the emergency? Please correct me if any of my assumptions above are incorrect.