This achievement is a little odd. First you need a warrior in your warband to be defeated, and suffer an injury of losing an arm or leg. This is something like a 5% chance per defeat, so you will probably get that during your playtime.
Second, pay the warrior's treatment and wages. The "treatment" will take 5 days, although no improvement will be made.
Third, re-equip the warrior to your warband, and then win a battle.
If the warrior is missing an arm, it won't be able to equip anything in that arm or climb, but it will get a bonus to its dodge chance. If it's missing a leg, it can't move as far or climb.
However, there seems to be a hidden requirement beyond winning the battle. Form my experience, it's one or more of these:
-Can't be done on a story mission
-Injured Warrior must not be defeated
-Injured warrior must deal final blow to one enemy. (ianroper1 got it without this)
-another warrior must lose a limb during the battle. (ianroper1 got it without this)
Let me know what it takes for you, so I can include that here, please.