I'm living in hotels right now and some hotels doesn't offer a lundry service. I've considered full service laundry where I can pick up my laundry ironed and ready to wear and also laundromats. I have yet to find a laundromat with drying rooms where I could put my business shirt and I'm not sure they'd survive a dryer.
Do not go to a laundromat but do go to a service where they clean and iron the shirts and return them to you in the hotel or keep them in the shop for you to collect.
That is what people do who do not stay in hotels with laundry services or who do not want to spend their time cleaning and ironing their shirts.
This service can be found at some laundromats, some hotels will tell you where to find them, dry cleaners will often offer the service.
And for the other places use your search abilities as every area will use different names.
In the USA, most hotels above the very lowest-price have an ironing board and iron in each room, or will loan you one on request. I have hand-washed many shirts in the room's sink, left them to drip until mostly dry, then ironed them myself. I did the same in Germany once, now that I think about it.
I solve this by washing my clothes in the bath, and drying them in the cupboard. The difficult bit is rinsing them, using a detergent designed for wool (eg. woolite) helps.