What is the best way to mirror an ASPState database?
Mirroring tempdb, that's a no. Have you checked Step-by-step SQL Mirroring the ASPState database?


  If the -sstype:p parameter is not specified, then by default, the
  ASP.NET sessions will be put into the TempDB database and not the
  ASPState database. This confused me for a while. This is fine for a
  normal non-mirrored environment because the ASP sessions will be
  cleared if the server is restarted. But, this is not fine for the
  mirrored environment, because we want to mirror the ASP session data
  itself, not just the Stored Procedures! Also, mirroring is not
  possible on the TempDB database. The -sstype:p parameter makes ASP.NET
  install the session tables into the ASPState database, and they'll be
  persisted if the server is restarted. This is exactly the behaviour we
  want for a mirrored environment.