I've been having the same problem for months. iPad 2 with iOS 8.3. The problem seems to be that the Mail app gags if there are too many messages. Go to Settings/General/Usage and look for the Storage block. Tap Manage Storage and you'll get a list of the storage used by, well, every app on your device. Mail will probably be near the top. When I looked at this earlier today the Mail app was taking 5.1 gb. I could not launch mail. And I could not send e-mail directly from Tweetbot. Here's what I did.
First, get a record of all the settings and passwords for your e-mail accounts. Make sure these are correct. Next, log on to your busiest accounts through webmail. Delete most of the messages on the server. (My ISP has an entertaining option called "Bypass Trash" that just deletes the messages immediately.) Remember, the Mail app is kind of stupid. It will try to download every message on the server. Hence, reduce that number.
Now comes the fun part. Go back to your iPad and select Settings/Mail, Contacts, Calendars. Pick one of those busy accounts and tap it. You're going to tap Delete Account, but first take a few screen captures (Home-Power) of the account settings. (Most accounts have multiple screens for setup. Don't be shy about the number of screen caps. You can delete them when you're done.)
Got the settings? Good. Take a deep breath and tap Delete Account. You'll have to confirm that in the dialog box that opens.
Do this for all of your busy accounts.
The problem seems to be that the only way to reduce the amount of storage Mail uses is to actually delete the account, thereby deleting the messages. My Mail usage is now a much more manageable 2.9 gb. (I have quite a few e-mail accounts some of which are very busy.)
As far as I'm concerned this is a bug in Mail. Other e-mail clients I use include options for compacting the mail database. Still waiting for Apple to catch up.
Hope this helps,
Tony Lima
in Apple's general neighborhood