Why does my keyboard become sluggish & keys don't respond to touch on my iPad 2?

In one app or all apps ? Have you tried a reset to see if that improves things ? Press and hold both the sleep and home buttons for about 10 to 15 seconds (ignore the red slider), after which the Apple logo should appear - you won't lose any content, it's the iPad equivalent of a reboot.

You could also try closing all apps completely via the taskbar and then see if that helps : from the home screen (i.e. not with any app 'open' on-screen) double-click the home button to bring up the taskbar, then press and hold any of the apps on the taskbar for a couple of seconds or so until they start shaking, then press the '-' in the top left of each app to close them, and touch any part of the screen above the taskbar so as to stop the shaking and close the taskbar.

Try the really simple basic things and see if they help at all.

Restart the iPad by holding down on the sleep button until the red slider appears and then slide to shut off. To power up hold the sleep button until the Apple logo appears and let go of the button.

Reset the iPad by holding down on the sleep and home buttons at the same time for about 10-15 seconds until the Apple Logo appears - ignore the red slider - let go of the buttons.

Quit all apps and restart. Go to the home screen first by tapping the home button. Double tap the home button and the task bar will appear with all of your recent/open apps displayed at the bottom. Tap and hold down on any app icon until it begins to wiggle. Tap the minus ➖ sign in the upper left corner of the app that you want to close. Restart the iPad.

That's great. 🙂