If I'm not mistaken about some detail of English grammar (not one of my strong points exactly), "She has stopped to run every day" should be accepted as well.
I am not a native speaker, so I may be wrong, but I remember reading that stop + doing something means that you ceased to do something while stop + to do means that you are doing a pause to do something else. E.g. I stopped smoking = I gave up the habit of smoking. I stopped to smoke. = I was going somewhere and stopped to have a cigarette. If so, "she has stopped to run" is not the same as "she stopped running".
She has stopped to run every day. = She has stopped (something) in order to run every day.
I'm not a native speaker, but In Norwegian I think that would be: Hun har sluttet for å løpe hver dag.
That is: "in order to" = "for å". Of course, this requires you to recognize when an English speaker is using "to" as a short form of "in order to"!