I'm thinking of using the nursing room in the washroom to straighten my hair
Why don't you use the mirror in the washroom to straighten up your hair?
The washroom doesn't have a mirror
Wanting to look presentable at work is understandable. If a communal area has a mirror you need then be sure to be respectful of others in the area. That way everyone can have their fair amount of time using what they need.
I think I'm going to go straight to a manager and complain about it.
I think if you take you complaint you might be able to get new mirrors installed. Until that happens please share or try to put a mirror in your room where you can get ready in peace.