Can this be written without "si? I thought the "sí" was for emphasis so I translated it as "He has demonstrated that he does like her" but it was marked wrong because of the word "does". In English we can say "He has demonstrated that he does like her" or "He has demonstrated that he likes her" where the word "does" just give more emphasis.
Good question, after some review I think I can answer. "Showed" is past tense, "shown" is past perfect tense. The difference being that in "regular" past tense there is no auxiliary verb, in past perfect there is. In this example that would be the "has". But both sentences really mean the same thing, "he has shown" or "he showed". Since the Duo example contains "ha"(has) they probably are only looking for it to be answered in the past perfect tense. But in conversation your response would certainly be understood by native speakers, and sometimes used by them as well. Some verbs do not even have a special form for the past perfect, but "show" does. For the ones that don't it is fine to form the sentence in the way you have done. I hope this has helped...
I think your answer is correct and should be accepted. "La" most certainly can mean "her." In my opinion, the "does love " should not change the meaning in Spanish, especially with the "sí"; it is the more emphatic in English. "Demostrado" can mean "demonstrated" also. Please report it if you come across it again. It helps all of us.