Success! I am on my third iPhone 5s. I have tried everything to fix the TID problem. LIke others, newly enrolled scans would start to deteriorate after 2 hours and fail completely within 8 to 24. I gave up. Then, a few weeks ago I learned from Joe_Fo that the sleep button in Settings/Touch ID could be used to expand the dwindling recognizable area, as kindly explained by Joe:
1. Unlock with known area, then hit sleep.
2. Slightly reposition finger moving away from known area, but not so far that you move completely off of it, maybe a little less than half the home button diameter. A slow creep is what you want to do.
3. Rinse and repeat step 2, slowly moving towards area that was not recognized after the initial enrollment.
"A slow creep" was key. I took about 2 minutes a day for about a week, slowly expanding the recognizable area - on one finger only, as much as possible. For some reason, this was far less aggravating than constantly deleting and re-enrolling a new print. I did it during down time - it was even kind of fun, like a game, to see how much of my print I could get the scanner to read. (Maybe this is what others have done all along but I was never aware of this procedure).
Anyway, after doing this every day for a week or so, TID now works great. It works instantly, nearly every time and has been doing so for over a week with no additional scanning. The difference is amazing. I can finally use this feature as it was intended.
Here's one thing, though. The weather has warmed considerably here in the NE. My skin is in significantly better condition than it was a month or two ago - not that it was that bad. But, it could be making a difference. However, my skin condition back in late October and November, when I got the first phone, was not bad and TID failed miserably then.