How to set audiobooks to repeat on iPhone
Hmmm, I'd never noticed that (but then again, I wouldn't normally think to put an audio book on repeat).

If the main thing you want to achieve is to be able to play your audio book on your iPhone, and you want the ability to repeat them, then you can still actually play them using the Music iOS app.

To do this:


Within iTunes on your Mac or PC, add your audiobooks (they can be from any source, including Audible) to a playlist
Sync that playlist to your iPhone 
Open the Music app on your iPhone
Select the playlist
Play any audiobook it contains (and use the repeat function)


Hope this helps!

[EDIT]

This answer is being edited to cover a couple of scenarios that may affect the above steps working. Basically, with the latest versions of iOS and iTunes, plus the onset of Apple Music, the above instructions may not work for some people.

If you are in this situation, please use the following steps instead:


Within iTunes on your Mac or PC, select the audiobook you want to play in the Music app on your iPhone/iPad/iPod.
Richt-click on the audiobook and choose Get Info from the context menu that appears.
Click on the Options tab.
At the top for Media Kind you should see Audiobook selected. Change this to Music instead. (Note, if you'd like to listen to your Audiobook using the Podcasts app instead, then select Podcast instead of Music. However, since the OP is wanting to use the 'repeat' function, we need to use the Music app.)
Ensure that Remember playback position is selected.
Click OK.
Now add your Audiobook to a playlist (perhaps one called Audiobooks).
Sync that playlist to your iPhone.
Open the Music app on your iPhone.
Select the playlist.
Play the audiobook and use the repeat function of the Music app.


IMPORTANT NOTES

If you are using Apple Music and iCloud Music Library is switched on in your device, you must also do this step: 


After Step 6 and before Step 7: Right-click on the Audiobook file and select Add to iCloud Music Library. 
This additional step can actually take anywhere from a few hours to days (not to upload, but for the iCloud music servers to process the file). So, while you can continue with Steps 7 to 11 above, the file won't be available on your device until the file has actually finished processing. The way to tell this is to check that the dotted cloud icon next to the audiobook listed in iTunes is no longer a dotted line to indicate it is waiting.
And yes, as far as I am concerned Apple Music and the iCloud Music Library have 'broken' the way many people want to use playlists - but that's another story!