Does the account you are using have enough permissions to use Database Mail? If possible, please try to send a mail using a sysadmin account.

Use (parts of) the code below to get configurion settings regarding Database Mail:

USE msdb ;

GO

-- show information about several Database Mail related settings

SELECT is_broker_enabled FROM sys.databases WHERE name = 'msdb' -- value 1 indicates enabled

EXEC msdb.sys.sp_helprolemember 'DatabaseMailUserRole' --- check members of the DatabaseMailUserRole

EXEC msdb.dbo.sysmail_help_principalprofile_sp -- lists information about associations between Database Mail profiles and database principals

EXECUTE dbo.sysmail_help_status_sp ; --- check status of Database Mail

--EXECUTE dbo.sysmail_start_sp --- start Database Mail in a mail host database

EXEC msdb.dbo.sysmail_help_queue_sp @queue_type = 'mail' -- list the state of the mail or status queues

-- show a list of all mail-items send past week

SELECT

sysmail_allitems.mailitem_id

, sent_status

, recipients

, subject, body

, send_request_date

, send_request_user

, sent_date

, sysmail_allitems.last_mod_date

, sysmail_event_log.event_type

, sysmail_event_log.description

FROM msdb.dbo.sysmail_allitems

LEFT OUTER JOIN msdb.dbo.sysmail_event_log

ON sysmail_allitems.mailitem_id = sysmail_event_log.mailitem_id

where send_request_date > dateadd(dd, -7, getdate())

and sent_status = 'failed'

order by

send_request_date desc