Here's an idea, but I don't recommend it for anything other than DIRE EMERGENCIES. I've never tested it. You absolutely should NOT use this as a standard data recovery technique. Improve your backup schedule instead, and you'll avoid the problem entirely.
1. Restore the backup to a new database with a different name. 2. Detach the new database. 3. Delete the new LDF file and copy the original LDF file into its place. 4. Attach the new DB using the new MDF and old LDF files.
This probably will NOT work, and it won't work unless your database was in full recovery mode AND you never backed up the log for the entire month. If the DB was in simple recovery, forget it, that data is gone.
If you attached "any old MDF file" would you be able to do one final Transaction Log Backup? (Assuming: all the Tlog stuff is in the LDF file, and it can be Backed up without referencing, per se, the MDF file - does that sound likely?)
Then restore the last-month's-full-backup followed by any previous TLog backups (since the last full backup) and then finally the just-saved-TLog-backup.
But as Rob said the database would have to be in FULL recovery model
Its quite difficult to recover data without mdf file. As you are saying that you have the ldf file. ldf file is the log file while mdf file is the master database file. Your whole data is saved in mdf file. I am not sure that you recover your data but try the suggestion given by others.