;)

The sentences are fine grammatically; that's exactly the way semicolons should be used. :-) If there's a problem, it's that the sentence is kind of distracting in its almost poetic, sing-songy tone. I'm not sure that the parallel construction of those three sentences really contributes anything. Since conciseness is a concern, I'd experiment with eliminating the parallel sentences and tightening up the paragraph. (The first and third sentences seem to be kind of redundant, for instance -- aren't "support," "comfort," "reassurance," and "encouragement" all pretty much the same thing? And the reference to the physical struggle doesn't really seem to be resolved.)

Your sister is correct.

Semicolons can be used as a conjunction, like "and," "but," or "or."

Sarah - no no no. The previous posters mean well, I suppose, but they could not be more wrong.

A semicolon tells the reader that the phrases on both side are different ways of saying the same thing. It's purpose is to help the reader take the time to put the information in a richer context rather than taking it at face value.

A semicolon is not a conjunction. If she is dealing with older doctors, then she will not do well on this part of the application. They will notice not only the careless punctuation, but also that she thinks "reassurance and encouragement" are appropriate for someone who was struggling physically.

On the plus side, because this application has a character-limit box for the "essay" is seems likely that it won't be read closely. They are probably really looking at grades, schools, and work experience.

A semi-colon is not only used to separate two like, albeit stylistically different, thoughts but also to separate two related thoughts/sentences, ie. the second follows from the first.
The usage of the semi-colon in your example is correct but repetitive.

Excellent job 'Not Kidding'!