The first thing to do is to take the external housing apart and see if there's anything obviously wrong - lose cable, dead giraffe across the interface pins, that kind of thing.
If that doesn't yield anything you are then faced with the 'nuclear' option - no, not central banks monetising government debt, but opening up the drive itself. There is a slight possibility that the head will have some contaminant that a cleaning tape won't remove, which you will be able to spot and carefully remove with a cotton-bud dipped in isopropyl alcohol, or there is a foreign object blocking the tape path.
Unfortunately, with the exception of high end devices like IBM's 3592 series, modern tape drives just aren't built to last. Manufacturers have cottoned on to the fact that they become obsolete in terms of their capacity in the space of a year or two, and have lowered design standards to match.
