How do you feel comfortable about failure and develop coping mechanisms to handle the emotional backlash? I.e. is reading relevant material helpful or is seeking out a mentor whom you feel secure with or something completely creative?
failure resilience career reading mentorship

I will just take it as a valuable lesson and seek better resolution (or advice), this will help assure you to overcome any possible situation that might appear in the near future.

seeking a mentor will will be a good option because your mentor will get you ready for almost everything you need to know and what you might expect,
learn from what happened and that will make you stronger, reading is good and seeking other mentors will be a good option as well.

Failure usually means it doesn't meet your desire results but some results are indeed generated. I always find deep for all the small wins I get from that failure and be grateful of it. Appreciation always makes you happy. That way your down time will be minimize and you will be back at your productive cycle.

First things first - failures are bound to happen as humans are fallible. Personally, I'd try to not beat myself up over it to the point where I shoulder the entire guilt and burden of the consequences - even after the matter has been resolved.

I would say do what works best for you to cope with that failure (of course, no self-harming activities) for just that moment. Example: I like to watch a nice short comedy on Netflix just to get my mind off things. Once I have a clearer mind, I would reflect upon my experience and note down my reactions, the external factors involved etc. Some people might do journalling, some might want to talk it out with a though partner or mentor - the point is to take stock of whatever that has happened that led to such experience and use it as a learning moment, not as a wholesale definitive moment of what you are as a person (ie. a failure).