Running a query within a scheduled job
As per the code in your comment:

USE msdb 

EXEC sp_send_dbmail 
    @profile_name = 'SQL Mail', 
    @recipients = 'vince.chan@ufa.com', 
    @subject = 'T-SQL Query Result', 
    @body = 'The result from SELECT is appended below.', 
    @query = '(include script from above)'


The reason this won't work is the same reason why this wouldn't work:

declare @query varchar(2048);

set @query = 'Select TOP 10 
      s.database_name, 
      m.physical_device_name,
      CAST(DATEDIFF(second, s.backup_start_date,s.backup_finish_date) AS VARCHAR(100)) + ' ' + 'Seconds' TimeTaken,s.backup_start_date,
      CASE s.[type]
      WHEN 'D' THEN 'Full'
      WHEN 'I' THEN 'Differential'
      WHEN 'L' THEN 'Transaction Log'
      END AS BackupType,
      s.server_name,
      s.recovery_model
 FROM msdb.dbo.backupset s
 INNER JOIN msdb.dbo.backupmediafamily m ON s.media_set_id = m.media_set_id 
 ORDER BY backup_start_date DESC, backup_finish_date
 GO';

 exec (@query);


You're not taking into account the single quotes in the query string.  Something like this should be what you need:

declare @query varchar(2048);

set @query = 'Select TOP 10 
      s.database_name, 
      m.physical_device_name,
      CAST(DATEDIFF(second, s.backup_start_date,s.backup_finish_date) AS VARCHAR(100)) + '' '' + ''Seconds'' TimeTaken,s.backup_start_date,
      CASE s.[type]
      WHEN ''D'' THEN ''Full''
      WHEN ''I'' THEN ''Differential''
      WHEN ''L'' THEN ''Transaction Log''
      END AS BackupType,
      s.server_name,
      s.recovery_model
 FROM msdb.dbo.backupset s
 INNER JOIN msdb.dbo.backupmediafamily m ON s.media_set_id = m.media_set_id 
 ORDER BY backup_start_date DESC, backup_finish_date';

 exec (@query);


Therefore, your constructed query string should be:

'Select TOP 10 
      s.database_name, 
      m.physical_device_name,
      CAST(DATEDIFF(second, s.backup_start_date,s.backup_finish_date) AS VARCHAR(100)) + '' '' + ''Seconds'' TimeTaken,s.backup_start_date,
      CASE s.[type]
      WHEN ''D'' THEN ''Full''
      WHEN ''I'' THEN ''Differential''
      WHEN ''L'' THEN ''Transaction Log''
      END AS BackupType,
      s.server_name,
      s.recovery_model
 FROM msdb.dbo.backupset s
 INNER JOIN msdb.dbo.backupmediafamily m ON s.media_set_id = m.media_set_id 
 ORDER BY backup_start_date DESC, backup_finish_date'


You need to use two single quotes in a string that contains a single quote.