Here's what I would do in that situation. Uninstall the program again. Run bleachbit user mode and root mode. Reboot, then do a search for all transmission files. Delete whatever you find. These are the directories where you may still have transmission files left after an uninstall:
~/.cache/transmission/ (~ = home directory, .cache is a hidden directory)
/usr/bin/transmission-* (directory is /usr/bin , files are named transmission-*)
Don't worry about others, as they're icons, messages and docs. Do not delete /usr/share/bleachbit/cleaners/transmission.xml. Open Synaptic, Reload, Mark All Upgrades, Apply. Then, reinstall transmission again.
If you still have problems, post back what desktop you're using, as well as the results of the following commands from a terminal: