You shouldn't need to re-enter your game code again, as you have already registered your game through Origin and should be ready to download and install.
You will need to make sure you are using the same Origin account you either bought the game with or registered it to.
However, if the game is not showing up in the Games Library try the following things:
- Restart Origin
- Log out and back in, using using the email address and username you used to purchase or register the game with.
- Make sure it's not under Hidden Games
- Restart your router/modem
- Uninstall/reinstall Origin
https://help.ea.com/en-us/help/origin/origin/manually-uninstalling-origin/
If the above isn't helping, contact EA's Customer Service:
1. Click on the following link: https://help.ea.com/en/contact-us/ (once there make sure that your country is correctly selected in top left hand corner of Help.EA.com)
2. Find the affected game - in this case select either Origin or Sims 4
3. Select your platform
4. Select topic and issue.
5. Select a contact option. Please note that available contact options do depend on your country settings, time of day and the topic/issue you choose.