ios 15.5 won’t install

If you have trouble installing an update, I would reccommend that you update via a pc/Mac

See this instructions if you're running a mac with macOS Catalina or later

Update your iPhone, iPad or iPod touch with the Finder – Apple Support (UK)

See this instructions if you're running a mac with macOS Mojave or earlier, or if you're running a Windows PC

Update your iPhone, iPad or iPod touch with iTunes – Apple Support (UK)

need more storage for update?

If you need more space for an update on your iPhone, iPad, or iPod touch - Apple Support

issues?

Updating iOS: common issues and solutions - Apple Community

Are you getting some sort of error?