Will my iPhone 5S update to the iOS 11

If you're sure that your iphone is a 5s model (not 5 or 5c) the device can be updated to the latest iOS version (11.4.1).

Read this support guide:

Get help with over-the-air iOS updates - Apple Support

If you still can't update, do it connecting your device to a computer running the latest iTunes version (now 12.8).

Update your iPhone, iPad, or iPod touch - Apple Support

If you don't have a computer, borrow one from your friends, or make an appointment at your most convenbient Apple Store or Authorized Service Provider. They will help you.

Apple - Find Locations - Location Selector

Regards

Giulio

ProustGiulio has provided excellent advice, but did you try restarting the phone and restarting your router and then try again?