Let's examine this in an empirical way, with the objective of building intuition about the effects of sample size and differences between RSSs. We'll simulate many samples of size $N=(5128-57) = 5085$ from a Normal distribution with mean $0$ and variance $\sigma_1^2 = 17283/5085 = 3.399$, calculate $2k + N\log(\text{RSS}/N)$, plot a histogram. The numbers have been chosen to match those from the model 1 results. We'll repeat the exercise with numbers chosen to match the model 2 results, and, by comparing where the two histograms lie, we'll be able to see whether the two AICs really have any chance of being different due to randomness or not.
N <- c(5085,5066)
k <- c(57,76)
s2 <- c(3.399, 1.997)
aic_res <- rep(0,20000)
for (j in 1:2) {
shift <- (j-1)*10000
for (i in 1:10000) {
x <- rnorm(N[j], 0, sqrt(s2[j]))
aic_res[i+shift] <- 2*k[j] + N[j]*log(sum(x*x)/N[j])
}
}
hist(aic_res, breaks=25,
main="Histogram of sample AICs",
xlab="Model 1,2 AIC")
leading to:

We can see there is no chance (in the practical sense of the word) that the difference between these two AICs is due to chance (in the statistical sense of the word.) The gap between the two AICs is several times the width of the histograms themselves.
How big a $k$ would be needed for these two histograms to more-or-less align? An easy calculation: $(6344.6-3635.5)/2 = 1354$.
So, what have you missed? The large sample size combined with the large difference between the two RSSs is such that a mere 19 extra parameters is, by itself, a small factor in the overall AIC calculation. There are two ways to look at this: how small a sample size would we need in order for the two histograms to more-or-less overlap, and how small a difference in the observed RSS values would we need in order for the two histograms to more-or-less overlap given the original sample size.
In the first case, reducing the sample sizes to $(169, 150)$ gives us the following plot:

which still would favor model 2, but not by much.
In the second case, a quick series of calculations shows us that the two AICs will be roughly the same if the RSS from model 2 is $17155.4$, an 0.74% decrease from the model 1 RSS of $17283$. Given that the actual decrease is far greater than that, it's clear that Model 2 is a better one.
These experiments were performed with Normal distributions, and other distributions would give you different results - for example using a $t(3)$ distribution would cause the histograms to spread out a little - but, given the magnitude of the differences between RSS and the large sample size, the qualitative results wouldn't change.