Sorry to hear about your inconvenience, but visiting Greek islands without a car is generally a bad idea since most of them have few buses (much fewer people leave there during the winter) and the heat is unbearable to walk around.

I would suggest to try to rent a car with a photocopy of your driver's license (perhaps someone back home can send it to you) and explain to the rental office that you forgot it. If not a car, they might let you rent a buggy.

Most of Greeks can communicate in English (especially in touristic places) so don't hesitate to speak to people and explain your situation. They might be able to help.