Orrick has a 500 word cover letter as well as asking "Why have you decided to pursue a career as a commercial lawyer working in an international law firm? What factors and influences have affected your decision?" and "Why specifically are you interested in training at
Orrick? What makes us different to the other firms you have applied for?" so what acc goes in the cover letter? I normally cover why commercial law and why the firm in my cover letters with why me woven in with them. Should I just talk about my transferable skills and experiences?
I think that you should still use the usual structure for the cover letter, so: WHY LAW, WHY THE FIRM AND WHY YOU. However, I would keep the why law and why the firm section shorter and focus more on why you as you suggested. You want to give them experiences, opportunities, prizes and awards you have won that are able to show how successful you would be on as a trainee solicitor at the firm, and the skills you possess.
I would try to emphasise what I learned from my past experiences too and link them to how you could use these to succeed at the firm. Additionally, whenever you are talking about why the firm in the cover letter, try to mention 1 or 2 reasons you have not mentioned elsewhere to showcase your research about the firm. You want to give them a thorough perspective of why you want to work at the firm. The cover letter is just another opportunity to do that, combined with the other questions on the application.
I hope this is helpful and good luck!