If you need apps for that iPad, you can try to download the latest app from the iOS App Store and if the developer has left an older, compatible version on Apple's app servers, then you will receive a small window pop-up prompt that states that there is an older version of the app that is compatible with your device. Would you like to download this version now.
Tap the OK button just below the prompt.
FYI,
Not every app you attempt to download will have an iOS 5.1.1 compatible version. For older apps that may still have an older iOS version available, the lowest compatible iOS version of an app is iOS 4.2.1.
The search for iOS 5.1.1 compatible apps will be trial and error, but they are out there.
This website may help you locate the apps you can use with your original iPad 1
http://findoldapps.com/us/any_category/5.1.1/ipad/updated_date/desc/30/1
My 2010, iPad 1 has over 100 apps on it that are still functioning, but older versions of apps that I currently run on my newer iPad models.
Use another third party browser to access to YouTube Website.
Set the YouTube website to display the desktop site version of YouTube.
Don't use the original YouTube app, any longer (the old YouTube app looks like a TV) that version of the YouTube app doesn’t work any longer.
There was a newer YouTube app that was available for iOS 5.1.1 (it was a red TV screen shape that just says inside YouTube), that version of the YouTube app still works on iOS 5.1.1, but I am not sure if you can download this older version any longer or not.
You could try downloading the current version of the YouTube app to see if the app offers your 2010 iPad an older version to download.
There are many third party web browsers that may still supoort iOS 5.1.1.
Here are one's that I have used or still use and recommend. There maybe other third party web browsers that still have older versions compatible with iOS 5.1.1.
iCab
Perfect Browser
Dolphin Web browser
These may still have an older 5.1.1 version that you can download from the iOS App Store.
Good Luck to You!