It’s been a while since I listened to an audiobook on an iDevice, but you have tried clicking on the "… More " button at the bottom of the Music app and selected audiobooks, right? Or is it that it doesn’t even show up then?
Nope nothing, when you click the more button there is no audiobook option at all, apparently because it doesn’t think I havce ever bought one so the option isnt active.
I’d be surprised if that were the case, but maybe. Just in case, press Edit when you’re in that More section and see if Audiobooks has been deselected. You can choose what shows up in there and it’s more likely it’s just not shown by default.
If it has been selected and still doesn’t show up, then I can’t suggest much more than trying to redownload the audiobook.