I have NO IDEA why you cannot login into the App Store with your Apple ID and password.
I just logged into the App Store today, with my 2010 iPad, using my Apple ID and password, to download two older versions of apps I have and use on my newer iPad models.
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.
Good Luck!