Rymdelefant Before doing the following, go into Documents > Electronic Arts and rename the Sims 4 folder to something that doesn't include "Sims 4." This folder contains saves and other user data, and renaming it will prevent the apps I'm listing from flagging it for deletion.
Please download Revo Uninstaller (the free version is fine) and see whether it can find Sims 4 on your computer. If it can, let Revo uninstall it. Either way, use Revo to uninstall the EA App, and choose the Intensive scanning method.
Restart your computer, reinstall the App, and try to install Sims 4 into a new folder that you create for this purpose. The folder should be on the root level of the drive of your choice, for example C:\Games is fine, but not C:\Program Files\Games.
If this doesn't help, try using Piriform's CCleaner (the free version is fine as well) to clean out any leftover Sims 4 data. This would include registry entries and other hidden files that may not have been removed when uninstalling the game, especially if Revo can't find it. Then use Revo to uninstall the EA App again, restart again, and try to install Sims 4 into a new folder.