Reset, hold both home and power buttons until the iPhone restarts itself, ignore the "slide to power off", let the iPhone restart itself. Usually takes about 10 seconds of holding both buttons. See if better. If not, tap Settings App > General > Reset > Reset All Settings. See if starting over helps. If still problem, Restore iPhone Firmware with iTunes on computer. See if better, it should be after Restore, because Restore is more thorough than the Update which started the problem.
I tried ever suggested fix, including multiple restores via iTunes and even the firmware restore. Nothing worked.
Worse yet, my wife's phone stopped allowing for capital letters. The shift key stopped working.
While typing emails or texts holding the phone vertically, certain letters didn't work. The same problem (with different keys) happened when holding the phone horizontally.
I finally made an appointment with the Genius Bar at a nearby Apple store. The guy tried everything he knew, to no avail, and then uttered this line: "I've done more than 3,000 iPhone appointments since I started working here and I've never seen this problem before."
He gave up and just replaced the phone with a new one.
Our warranty had 21 days to spare, so we just made it . Whew!
Thanks for the suggestions.