I think one of the best ways to approach this kind of sentence construction in English is to change the question form of the sentence into the declarative form (though there are exceptions). Therefore, "What does the lady think has happened to Lorna?" should be changed to "The lady think 'what' (something) has happened to Lorna'." As you can see, the "what...Lorna" is a noun clause here, which already acts as a noun. So in this case, it is not grammatical to add the word "that" here or it would look like this, "The lady think that what has happened to Lorna," which does not make much sense in English grammar.
However, it is worth pointing out that the word "that" that introduces a noun clause is usually omitted, as in "She told me (that) ...." And interestingly, when it comes to the conjunction "so...(that)" the word "that" can be left out as well.