I found some information on combat

here. No idea how up-to-date that is. First strikes basically give you a chance to deal some extra unanswered damage -- basically a combat round where the first striker

*may* deal damage, but if they fail then they don't take any damage in response. The amount of damage you deal depends on the ratio of your strength to the defender's strength.

Say the archers you were attacking had strength 4.5 (CG1 promotion) and your CKNs have strength 6. According to the link above, your ratio is 1.33~, which means that you have a ~57% chance of winning each combat round, and each time you do, you deal 18 damage. If you have 2 more first strikes than the archers do then that equates to dealing ~20 damage at the start of the fight, on average (.57 * 18 * 2). That's about a fifth of the archer's base strength; a decent help but nothing amazing.

If you're fighting an unpromoted war elephant (strength 8), then your ratio is .75, which gives you a ~43% chance of winning any given combat round for 17 damage. Say you get 4 first strikes, that's ~29 damage at the start. But for the rest of the fight, with your much lower ratio you're going to have a tough time of it. Things get worse if it's promoted, of course.

This is why first strikes are generally mostly useful if you out-tech your opponent. A machinegun with 3 first strikes vs. an attacking unpromoted maceman will have a ratio of 2.25, giving a success rate of 69% and damage per success of 29 -- that's 60 damage! Couple that with the greater likelihood of success in normal combat rounds and the machinegun will come out of the fight almost unscathed.

If I recall correctly, the animations played during combats basically are intended to create an "interesting" exchange, and generally have next to nothing to do with how the combat actually plays out. I turned them off after watching my 20th iteration of "attacker kills 2 of defender's 3 guys, then remaining defender one-shots all attackers". It gets pretty recognizable.

Generally speaking, you're better off improving your ratio by taking Combat promos (or specific counter-unit promos like Cover or Shock) than you are taking Drill.