Is there any software to compute and display furigana completely offline?
It doesn't have to be integrated with any existing app. A new standalone app I type into is fine.
I'm a little confused here. You want an app that you can type kanji into and get the kana in return. But the only way to type kanji into the iPhone is by typing it phonetically, which gives you the kana for free, essentially.
There are a number of self-contained Japanese dictionary apps for the iPhone. Most of them use the EDICT dictionary. I've got one that can import EPWING-formatted dictionaries, and have imported Kenkyusha's New Japanese-English Dictionary into it.
This seems like the closest to what you want. I think. I get the impression that it has the dictionaries built-in, but I could be wrong.
You have several apps available on the App store now - just search for Furigana.
I have a friend that likes this one, but YMMV:
It does not work offline, so it doesn't meet your requirements unless you fail to find any offline software and can relax that constraint.