This boot-info script
http://sourceforge.net/projects/bootinfoscript/ will help you determine which partitions are used by each OS.
Then you can simply mount the "mint" partition from inside Ubuntu and remove all the files you don't want. If you have HOME directory files that you'd like to retain, copy them over to the HOME on Ubuntu first.
If you aren't already doing this, it is a really good idea to have 3 partitions for a Linux desktop.
1) The OS - usually /
2) HOME - /home
3) Swap
In the old days, we'd create partitions for /usr/local and /var too, then mount all the programs over the network for /usr to save space across 5-200 machines.
If you want to play with interesting file systems, you might want /boot to be a different partition too since booting isn't supported from every file system available. If you are using EXT4, this doesn't matter at all.
Sorry - that was probably more than you wanted.
To clean up the old Mint install from the grub menu, look for a grub howto on that. Be certain you are using the correct howto for your exact version of grub. I know it is confusing to me.
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