Battery drain and high cpu usage by e-mail app, what should I do?

I am currently running android 4.4.2 on my Nexus 7. Randomly my battery drains completely in a few hours. I noticed this week that when this is occurring, I checked which app is using the most battery and it is the e-mail app. I also noticed that I am getting an internal error when in the e-mail app. The battery drains even if it is not opened.
e-mail v6.1-937116

I'd update to 5.1.1 or at the very least 4.4.4
That may help with issues you're currently experiencing.

The basic difference between 4.4.2 and 4.4.4 is the OpenSSL fix - nothing to do with email or battery. (I'm still getting 2 days on my Note 3 on 4.4.2, so the version of Android isn't the problem.)

Install a different email app, and uninstall the one you're using. It could have gotten corrupted. Or maybe first try deleting your email account and see how the battery behaves for a few hours. If the app is still using a lot of battery, I'd uninstall it. (I'm assuming that you don't have it set to check for new email every 5 minutes or something like that. That will kill your battery fast. If it's an IMAP account, set it to never check. The server will notify the phone when there's email.)