Solution 1:

Usually in scenarios where the disk won't mount and you get that error message the computer is trying to run a fsck in the background, check activity monitor for a fsck_hfs or similarly named process and kill it, the disk will mount for you if you're lucky.

Solution 2:

When this has happened to me, I've used data recovery programs to recover the data. I would advise not to attempt any disk repair (such as Disk Utility or fsck) or other alterations until after you've recovered as much data as possible. The last time this happened was many years ago, and at that time I used a shareware program called DataRescue. I especially liked that it ran in trial mode and showed you what it could recover, and let you recover some of the data, and then, if you were happy, you paid and got a license and recovered everything. Since then, the program has a new company and I don't have experience with it, so I can't speak to the current version. If it still has similar terms, I'd suggest trying it.

If you don't want to try data recovery programs (which I would do first),an alternative if you can see the drive but the volume won't mount, would be to use dd to try and copy the data to a virtual drive. Use Disk Utility to create a sparse disk image, then dd to copy the sectors from the broken disk to the virtual one, then Disk utility to try and repair the virtual one. But I'd try data recovery software first.