The Google Home app is used on iPhone to detect and set up smart home devices including the Google Chromecast. Older Chromecast models, such as the first generation may have trouble during setup. There are several steps you can take to help correct the problem, and find the Chromecast in the Google Home app.
To troubleshoot the setup process, make sure the Chromecast has a flashing white LED when plugged into the TV and a power source. Otherwise reboot the Chromecast, or factory reset it by holding the Chromecast button for 20 seconds.