I’d write about your experiences as a cadet so far, qualification’s you’ve gained and would like to gain, and most importantly, why are YOU applying for promotion? Your staff may find it useful to know what makes you tick when assessing your suitability for a JNCO role.
Have a clear paragraph explaining your strengths and weaknesses (and what actions you plan to take in order to self-improve). Saying you have weaknesses is absolutely fine (unless it’s something extreme!), and it sounds a lot better to explain how you plan to fix them, rather than simply stating a long list of strengths. If you’re struggling with your strengths/weaknesses, speak to another cadet about it.
Regarding the ‘cocky’ side of things, perhaps read a draft of your letter to a fellow cadet or someone you trust, but a bit of self-confidence is never a bad thing!
Finally, I presume that you will have some sort of interview or meeting after applying, so anything you forget in your letter can always be mentioned then.
Best of luck!
Some useful threads:
Also, look up ACP 48, the JNCO course notes, if you’re unsure of what you’re aiming for in the role.