Two dozen kissing number approximations
Julia 0.6, 52 bytes



n-&gt;ceil(n&lt;8?3.7exp(.51n)-5.1:n&lt;24?9exp(.41n):196560)


Try it online!

How?

Machine learning! (Kinda. Maybe. Not really.)

First, from plotting the lower and upper limit data against N, and some manual trial and error, it seemed like an exponential function could fit well for N >= 8. After some attempts at manually finding this function, I resorted to using a parameter tuning function to tune \$ a * e^{b K} + c \$  (where K = 8 to 24) to find a, b, and c such that the expression gave values that lie within the correct range for each K (with successively smaller range of possible values and correspondingly higher precisions for a, b, c in different runs of the function). There were a few such set of values, though none of them could fit the N=24 case exactly, so I went with one that had 0 c and hardcoded the value for N=24.

For lower N, from 1 to 7, I was hoping to find some simpler expression or polynomial, but couldn't find any that fit. So back to fitting \$ a * e^{b K} + c \$ , this time for K = 1 to 7 (though I didn't really think an exponential would be a right fit in this case, based on visual plot trends). Thankfully, there were parameters a, b, c that could give correct values throughout the range here (at least to within a ceil call).