As the sibling comment already said "kai" (pronounced ke like in keg) just means "and". So it literally means 4 and 10 sides in greek. But I have often seen it written as τετραδεκάεδρο (tetradecahedron) in greek as well, so without the kai part. I'm not sure why it is 4 and 10 instead of 14 though. It would be more natural in greek that way (δεκατετράεδρο - decatetrahedron). Maybe it is for putting the distinctive part (4) first, or maybe it sounded more "poetic" like that to someone and then it stuck.
I think the three-and-ten, four-and-ten way of expressing numbers is primarily an ancient Greek thing. The modern numbers are expressed differently (δεκατέσσερα / dekatessara for fourteen, for example). In a lot of older European languages 11 and 12 behaved irregularly. You could argue that they do in English too (we don't have oneteen and twoteen).
I haven't read of any particular reason for this, but I'd posit that numbers up to twelve were more commonly used in everyday life, so shorter, irregular forms were easier to use and remember. Much like many of the irregular verb forms in spoken language happened because they were so commonly used.
The ancient Greek system also gave us triskaidekaphobia - the fear of the number 13.
Indeed, in ancient greek they put the "δέκα" (ten) part second. Τρία και δέκα (13), τέσσερα και δέκα (14), πέντε και δέκα (15), etc, but 11 and 12 was (and still is in modern greek) irregular, έντεκα (enteka) and δώδεκα (dodeka) respectively.
French has some weirdness to its counting. 11 - 16 is specific words, 17 - 19 is "ten {number}" (dix-sept, dix-huit, dix-neuf).
There's also some fun when you get to 70... which is soixante-dix (sixty ten)... and 80 which is quatre-vingts (four twenties)... and 90 as quatre-vingt-dix (four twenties ten).
Some dialects of French outside of France have changed how they count.
With Spanish being similar with distinct words for 11-15 with 16-19 being 10+number. Taking Spanish in high school made me consider that for the first time while never having thought about 11,12 being similar in English until that same thought process.
Latin has 1 to 10 as quite reasonable. 11 is "undecim" (one ten), and 12 is "duodecim" (two ten)... 16 is "sedecim" (six ten), 17 is "septendecim" (seven ten)...
18 is duodeviginti (two from twenty) and 19 is undeviginti (one from twenty).
Yes, there are lots weird remnants of the vigesimal system around in French. You can see it in names like the Quinze-Vingts hospital - literally "fifteen twenties" because it was designed to have 300 beds.
There were plenty of base-12 number systems in Europe, one way of counting to 12 is using your thumb to count the bones in the other 4 fingers, one hand for the 1's digit, the other for the 10's digit.
12 being divisible by 2,3,4,6; 10 being divisible by 2,5 -- means base 12 is easier to multiply and divide to reach whole numbers.
Ancient Greek spans several centuries of sound shifts and many dialects. It cannot easily be simplified into one specific pronunciation, particularly not one that is based on your specific dialect of English. Wiktionary has /kǎi̯/, /ˈkɛ/, /ˈcɛ/ and /ˈce/ for "καὶ".
You don't have to span several centuries to witness that. "και" is pronounced in several different ways in modern greek as well. But that's besides my original point.
