The iPad 2 is notorious for display issues, and a lot of the time the solution is to smack it on the back like you would when trying to burp a baby. I don’t believe this to be any type of problem with the digitiser, so a replacement of the glass would not help.
if smacking the back of the iPad doesn’t help, you could also try and remove the front glass and ensure the LCD cable is fully connected into the logic board (search for an iPad 2 screen replacement guide to learn how to disconnect and reconnect the LCD cable).
if all else fails, you could purchase a replacement LCD panel and install that. Being the iPad 2 is an old tablet, replacement parts aren’t too expensive.
Again, if you need to replace the LCD, follow the guide on this site for the best instructions on how to do so.