After trying many different ways I could not fix iPhoto. However I succeeded to import photos to iPhoto. The trick was to run Image Capture, import photos using it and selecting the destination to iPhoto. It worked like a charm!

Apple has a support page about this.

Most programs will show the original, unedited version of the image.

Images you import to your computer directly from your iOS device will show up as the original image in iPhoto, Aperture, and in the Finder.

To import edited photos to your computer

If you want to view the edited version of a photo on a Mac or PC that doesn’t automatically import the edited version:

1. AirDrop or email the photo to yourself.
2. Or, if you've enabled iCloud Photo Library, download it from the Photos web app at iCloud.com.

This will save a copy of the edited version on your computer, but you won't be able to revert to the original, unedited version.