Install the widget and see if you get something in a log as to why the TM won't start up.
http://pondini.org/TM/Troubleshooting.html
See A1..
But you may need to look in other logs because the software is failing completely to start it might not give a message at all.
I think all our troubleshooting is issue with TM but it has to have started.. if it refuses to start at all, sorry but I think you are going to need to use a clean install.
Of course TM is not the only backup software around.. there are plenty of excellent utilities for the Mac.. Pondini lists a few.
CCC, Superduper, Chronosync.
In some ways these have superior features.. particularly that they produce a full clone of the disk, rather than just a jumble of everything.
http://pondini.org/TM/Clones.html