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