'Undeliverable messages' may be because an address doesn't exist or is misconfigured etc. Normally these will be 'bounced' messages that look like duplicates of what you sent with more details.
Create a new message & try sending it to another email address, ideally an account you will get quick feedback from like a secondary gmail, yahoo or other free service that you have access to.
Can you use your email and password to sign into iCloud.com to try sending and receiving mail there? It will confirm your mail account is functioning on Apple's servers. You will need to use a computer - Apple do not allow iOS to access iCloud.com.
For what it is worth reboot the device to see if anything changes. If you use iTunes to backup you could backup the device then consider deleting the email account in the iCloud settings & then re-enable it. That should reset the iCloud settings if that is the issue.