An enlightening remark to begin with. External hard drives are nice but rather vulnarable and error-prone (compared with internal ones). So if it's a bad idea already to have data only on an internal drive (and not on a separate backup), it's a VERY bad idea to have data only on an external drive, especially if they are "very very important". You're sure to lose them then sooner or later.

That being said, I would recommend the following actions:
1. Open the enclosure, disconnect and reconnect the drive.
2. If that fails, mount it (as a slave for example) in your PC or an other enclosure.
3. If that fails also, it's time to either give up or call a professional data recovery company. But that won't be free!

Impossible to predict the outcome.

Hope this helps.