Images with the wrong mime type are unlikely to show up in Google Image search. Unless your images have an image mime type, they won't get indexed for image search.
As you note in your question, there are also usability problems with incorrect mime types. Nobody is going to want to use a page with broken images. While I doubt that Googlebot is going to carefully check the mime type of your images when evaluating the quality of the pages, Google does measure the usability of your page. If Google sees that users turn back from your site and click on other listings in the search results, your rankings will fall.