Here's your problem >>
its due to the fact that exchange was rolled out a few weeks before the new BE server was built and the exchange server ran out of storage space and some of the log's were deleted to make space.
>>> Your exchange is not mounting because you deleted log files.
Your backup is not running because your exchange is not mounted.
Please ** DO NOT ** remove the log files.
If you do - your exchange will not mount anymore.
Log files are cleared when an Exchange-aware backup solution, commits those transaction logs and then flushes it.
At this point - the options for you is Hard-repair of exchange.
STEPS:
a) Backup al 3 Dirty shutdown edb's.
b) Check shutdown status of your .EDB
eseutil /mh "Full Path to your EDB"
c) eseutil /p "Full Path to your EDB"
/p is hard-repair
d) eseutil /d "Full Path to your EDB"
to defrag
e) isinteg -s servername -fix -test alltests