IOS 6.1.3 upgrade has no airplay icon to mirror iPhone 5 to Apple TV.

It appears on my iphone 4 iOS 6.1.3 when I double tap home button and swipe all the way to the right.

Try a restart

Hold down the Sleep/Wake button and the home button together until the apple logo appears (ignore the ON/OFF slider) then let both buttons go and wait for phone to restart.

Out of curiosity, Costa, which AppleTV do you have? I have a first generation (big white/silver AppleTV) and have had trouble conecting with iOS 6.1.3. Interestingly, iTunes on my Macs is able to find it just fine - but my iOS devices cannot.

And yes, they're all on the same Wi-Fi network.

-M