Linux Mint 10 has been obsoleted and has reached end of life as of April 2012. Step by the repositories are being deleted for that release. You would be recommended to install a newer release to replace Linux Mint 10. Linux Mint 13 is the current long term support release, with support till April 2017. Linux Mint 14 is the latest release, with the newest software and support till April 2014. You can download either from here: http://www.linuxmint.com/oldreleases.php
. MATE or Cinnamon would probably be a best fit if you used the main edition (GNOME 2) of Linux Mint 10.
As for problems with the packages.linuxmint.com repository or current releases of Linux Mint, as detailed a mirror server is the solution for that. Please read the errors you are getting though on a current release of Linux Mint; if it is for something other than for packages.linuxmint.com, you have a different problem.