re: How difficult is it to get your kid into a military service academy

For USNA, probably start by talking to your area Blue & Gold officer.
LINK
I'd imagine West Point has a similar organization.

If your area person isn't responsive, there's probably other good info / people you can reach out to on the site I linked... and there's a place on the home page where you can request more info.

I assume class sizes are similar to when I went (early to mid 90s)... Each entering class was in the neighborhood of 1000 students at USNA itself, and another 200-300 at the Naval Academy Prep School in Newport, RI. I'll list the breakdown (per my memory, which might be suspect) so you can gauge the probabilities a little better:

- 2 congressional appointments... don't quite remember if that was per state or per Senator per state, or if the House of Reps could nominate... which I doubt off the top of my head. So, if 2 per state, that's 100 per year (or approx 10% of the incoming class of 1000); if 2 per Senator, 200 is approx 20% of incoming class... this is very much the basis of people posting you need to have an in with a politician. (edit to add: I see other people above my post saying congressmen can recommend more than two... maybe up to 10? ... I don't know... they might be right about that. Wasn't really a topic that came up between classmates after the first day or two.)

- Athletic scholarships... Football is probably about 80 students. Add in the various men & women basketball, baseball, swimming, track, etc., it was probably 200 or so athletic scholarships (another 20% or so).

- Prior enlisted. Secretary of the Navy can nominate / give appointments as well. I don't know if there's a cap, but was in the neighborhood of 80-100 (approx 10%) enlisted people when I went... this is how I got in. I was an enlisted Marine when I applied and got a SECNAV appointment. (If your kid really, really wants in to a service academy, this is one back door... though it's definitely a low probability gamble to enlist with the hopes of getting an appointment. If you do keep this option open, going enlisted Navy and getting into the nuclear submarine program is probably the best bet... I'd guess half of the prior enlisted contingent at USNA were from that community (had to be somewhat smart to be a Nuke).

- I might be missing another category, but think the rest of the class (very roughly 50% of the class) were called "at large" appointments... I don't know the application process (due to coming in through the enlisted ranks: I only remember filling out an application and getting 2 officer recommendations), but I suspect these students went through some sort of interview process beyond submitting the application with references... This is worth mentioning, because imho, I suspect congressmen are not interviewing many, if any, students directly, so if you aren't one of the lucky small number to get the congressional bid, there's still hope for an at large bid when the administrators look to fill in the rest of the class. (*** To this end, I recall one of my classmates mentioning working as a Blue & Gold officer and saying he helped interview prospective candidates... so, again, getting in good with your area's Blue & Gold officer is probably the easiest way to gain access to and make a favorable impression with someone who might have actual influence in the process.***)

The 200-300 students at Naval Academy Prep School (NAPS) were mostly athletes who didn't have a strong enough academic background to go straight to USNA and prior enlisted (like myself) who had been out of school for a couple years, a small contingent of Coast Guard Academy kids, and I think my year at NAPS had 2 or 3 kids who went on to the Merchant Marine Academy.