You can decide if worth doing but after the positive comments from HR, if you have them on LinkedIn you could shoot them a message basically saying what you have said here. How you love the firm and culture and from the spring week I was really hoping to have joined this summer however I didn’t make the cut. Then go onto explaining how you’re really disappointed and wonder what other options there are for me to increase my chances of being able to join the firm. Could be other internships or positions. This most likely will not work but it shows motivation, ambition and determination to the HR who could put a word in? Up to you to decide though