Free version (iOS App Store link) functions like a special notepad. You can type ancient (polytonic) Greek within the Keyman app and then export the result to another app. Keyman actually supports many other languages.
Pro version (iOS App Store link; $4.99 as of 2015-11-19) works as a system keyboard and can be used directly in any app.
I tested only the free version of Keyman.
Several keyboards for ancient Greek are available in the app. Scroll down to select Greek, Ancient. Then choose one of the listed keyboards. You can install multiple ones and switch between them. I tried a few; here are immediate thoughts (note that I didn’t read any manuals).
Greek Classical – Diacritic entry relatively easy (type after the letter; combinations of accent and breathing can be typed in either order after the letter. No separate key for final sigma; appears automatically if you type a space after a sigma. Deadkeys clearly distinguished from others. Access to roman letters in same keyboard, but one key at a time. Less familiar (to me) letter arrangement.
Greek Phonetic (Galaxie) – Diacritic entry relatively easy (type after the letter; combinations of accent and breathing can be typed in either order after the letter. Iota subscript typed by first typing the letter, then typing what looks like an ellipsis (…) key, then typing iota. Letter and diacritic arrangement least intuitive to me.
Greek Polytonic Unicode – Type diacritic before letter. Couldn’t figure out how to type both accent and breathing over same letter. No iota subscript?
If you want to try this app, I’d recommend starting with the “Greek Classical” keyboard (pictured below; dotted circles indicate diacritics that will combine with the previously typed character; typing the “Ltn” key will enter the next typed letter as roman, but note that the letters as displayed on the keyboard will not change).