You're able to access email accounts, such as iCloud, Gmail, Outlook and other, online, using a browser. To access iCloud, enter
www.icloud.com (and request the desktop site, if you use Safari and the software on your iPad is 8x), then login with the email address and password. You're able to send emails to your contacts from there as well, not only from the Mail app of your iDevice. That's what I meant.
I was asking if this works in case you tried.
Try the usual fixes, if you haven't done so:
Force close the Mail app - double tap the Home button to reveal the multitasking bar, slide the picture above the Mail app icon towards the top of the screen.
Reset your iPad - press and hold Home and Power button until the Apple logo appears, let go of both; the iPad will turn on on it's own.
Restart it - turn it off and on again.
Reset all settings in Settings - General - Reset. This will set everything you changed in the Settings back to default. You'll have to look at every setting afterwards and enable/disable what you need.
If this doesn't help, maybe restoring your iPad will solve the problem. Use iTunes to restore it. Reset your iPad afterwards, to remove minor glitches.