Hello Magali,
Delighted that you are planning a trip to Ireland & hope to include a visit to Northern Ireland whist here.
You will be glad to hear that there are no check points along our border, and so ID will not be required if visiting Northern Ireland.
I would advise however, that if you are hiring a car and driving across the border that you make sure you inform the hire company at the time of booking as failure to do so may mean your insurance would be invalid should anything happen whilst in Northern Ireland (assuming you are hiring a car in the Republic)
I am unsure what part of Donegal you are staying, but Derry~Londonderry is also very easily accessed by public transport from the main hubs of Donegal Town & Letterkenny; should you chose not to drive. The Walled City is steeped in history, dating back to St Columba founding a monastery in the city in 546AD. Come take a walk along the city walls, visit our Museums & Guildhall, take a walking tour through the Bogside and cross our Peace Bridge for some fantastic views of the city. The Causeway Coastal Route & Inishowen Peninsula are also quite easily accessed from the city, making it a good option for further exploration.
Happy Planning & we look forward to welcoming you to Northern Ireland soon.
Kind Regards
The Visit Derry Team