The easiest way without having to call support is to follow these steps:
1. Go to
https://account.lds.org/recovery?type=username
2. Click "I don't know my username"
3. Click "Membership Record Number" (you might have to get that from a clerk, or it's on a temple recommend if they have one)
4. Fill out the form with membership number and birthdate and click Recover
5. Once the username is displayed, make a note of it.
6. Go to
https://account.lds.org/recovery?type=password
7. Enter the username and click Recover
8. It should prompt you to reset the password
It looks like the new form when you click "Forgot your username?" on the sign-in page doesn't have the recover by membership number and birthdate option anymore, but the old one above probably still works.
If the above steps don't work, you might need to contact support.