Lost passwords are the worst. And it happens. After I got all 3 laptops working Apple suggested I wipe my Time Capsule clean and start over with fresh back ups rather than trying to link new computers to old back ups. Gulp.
Somewhere along the line we had to give the Time Capsule a new password and we did not write it down. I think we mistakenly thought we'd remember it. Right.
The following instructions work IF when you are asked if you want to save your password to your keychain you say yes.
Go to the magnifying glass in the far right corner of your screen.
Click on it.
Type the word "key".
You will see your keychains. Click on that.
You will see all your passwords. Click on the item you are looking for such as Rae's Time Capsule. Look at KIND and you will see the AirPort base station password.
Double click on that.
Check the box in front of Show Password.
It will ask you for a password. Use the one for YOUR computer.
Knowing this feels like pure magic.