The official YouTube app no longer supports iOS 5.1.1. Sideloading a modified IPA or applying legacy jailbreak tweaks are the only ways to restore functionality.
This is the most effective way to restore functionality to the original pre-installed or App Store YouTube app. youtube ipa for ios 511 verified
Getting YouTube to work on iOS 5.1.1 (most commonly for the iPad 1st Gen) requires specific "fixes" because Google long ago deprecated the APIs used by legacy apps. Simply installing an IPA file is usually not enough; you must also apply a server-side "repair" tweak via a jailbroken device. The official YouTube app no longer supports iOS 5