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.

When you get the message "disk needs repair" with amber flashing light from your Time Capsule, Please be sure to check that anther external hard drives but internal disk drive of TC is ok. Taking out it from TC is easy and checking it by Disk Utility from your MacOS laptop or desktop removes all the problem instantly before being suspicious about the internal hard drive of TC.

I was in a daze that I had to zero-out the internal hard drive mistakenly and then I found the error was being sent from one of external hard drives to TC.

This is a reference in favor of your convenience!