How to fix a Time Capsule disk in "Internal disk needs repair" status?
It is simple but non-obvious, as you have discovered...

There is an Apple KB article that states:

"Simply restart your AirPort Time Capsule. AirPort Time Capsule automatically performs a file system check on the internal drive during startup. If the AirPort Time Capsule detects an issue that it can resolve, it will repair the file system of the disk. If an issue is found that cannot be resolved, the AirPort Time Capsule's LED will flash amber, and AirPort Utility will state the issue. You should use the most current version of AirPort Utility."

You can find that here.

Failing that you may have to remove the disk from the airport (ifixit.com has instructions) put it in an external USB drive sled and attach it to your Mac for further repairs and diagnosis. If Apple's Disk Utility cant fix it then you may have to use a commercial utility on it like DiskWarrior or Drive genius.