The calendar new event page which you say should have a "Done" only has cancel or add buttons. Press either and what you have written disappears. No way to save calendar event that I can see
IOS – Adding calendar events on iPhone
calendariosiphone
Related Solutions
Tapping on an event should give you an edit button in the upper right. From there you scroll down to the very bottom and will see the option to delete the event(s).
delete event > 
Go to Settings and click iCloud and turn on Calendars.
If that doesn't work, while in your Calendar, look at the bottom - "Today," "Calendar," and "Inbox." Click on Calendars. Somehow my "Home" calendar (which is where I save most entries, as opposed to "Calendar" or "Work") was unchecked. I couldn't see any old or future entries, and when I added entries and hit save, the day would be empty. When I re-checked "Home," everything reappeared.
It just wasn't showing "Home" entries. I hope your problem is this simple because I know how you feel like you're losing your mind when something like that happens!
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Best Answer
The "Add" button is the one you want.
Your phone likely has multiple calendars. It seems like you are currently adding new events to a calendar that is not visible.
First open the calendars app and note the word "calendars" on the bottom center (directly above the home button). Tap the word to configure which calendars are "visible". The calendar that you are currently adding events to is not visible. You can make all calendars visible simultaneously and you should find the events that "disappeared" as you created them.
The second setting to be aware of is "Settings > Mail, Contacts, Calendars > default calendar" (currently the last entry on that screen). This determines which calendar new events are added.