I am sorry for the firm not contacting you with an update on your application, you might want to email them to ask for an update, reinforcing your interest in the firm.
However, I think that your cycle was very successful with 5 assessment centres. You should be so proud of yourself and I personally believe that this cycle made you understand that you are very strong at application stage so that is not your weakness. What seems to be a challenge is managing to convert these assessment centres into actual offers, which is a very daunting, difficult and sometimes outside of our control, so I would say to reflect on this, reflect on what you think you did wrong and what can you do to improve. Moreover, start practicing law firms interviews with family and friends is such a good way to gauge how you come across and ask them to be brutally honest with you so you can take the feedback on board and improve yourself.
When it comes to mentioning this experience, I personally would not mention it as I would want to focus on the positive aspects of myself. However, for questions around 'a time where you did not succeed or you failed at something, how did you learn from it' I think that this experience could be a good example to make. It shows that you are resilient, you learn from your mistakes so you are teachable and you are hard-working.
Good luck with your journey and congratulations for the progress you made thus far!