Couple of things:

SET NOCOUNT ON;

BEGIN

This needs to be at the beginning before you declare your variables.

Also, you're passing a string into xp_cmdshell that has "@PriorMonthStart" and "@PriorMonthEnd". You need to build the string in such a way that the values you want are passed, not the variable names:

SET @cmd =

'"SELECT ServerName as Server,

LoginName as FailedLogin,

ClientName as ComputerName,

LoginDate

FROM DBA.dbo.AuditDatabaseLogins

WHERE AuditType = ''Login Failed''

AND LoginDate between ''' + @PriorMonthStart + ''' and ''' + @PriorMonthEnd + '''

ORDER BY LoginDate" -o C:\AuditReports\FailedLogins.txt'