Best I can find is that the first known use of the term Lady Killer is in Je ne sçai quoi: or, a collection of letters, odes, &c. Never before published. By a Lady. 1769 written by Anne. B. Poyntz. Where it has essentially the same meaning as now.
Per: https://www.etymonline.com/search?q=killing%20%28adj.%29 the use of killing someone in that kind of a positive sense of "overwhelming someone" dates to the 1630s.
My American Heritage Dictionary lists under Kill the meanings:
b. To deprive of life: The Black Death was a disease that killed millions.
2. To put an end to; extinguish: The rain killed our plans for a picnic.
3a. To destroy a vitally essential quality in: Too much garlic killed the taste of the meat.
b. To cause to cease operating; turn off: killed the motor.
c. To tire out completely; exhaust: "The trip to work, and the boredom and nervousness of jobs, kills men" (Jimmy Breslin).
4. To pass (time) in aimless activity: killed a few hours before the flight by sightseeing.
5. To consume entirely; finish off: kill a bottle of brandy.
6. Sports To prevent a hockey team on a power play from scoring during (a penalty).
7. To cause extreme pain or discomfort to: My shoes are killing me.
8. To mark for deletion; rule out: killed the story.
9. To thwart passage of; veto: kill a congressional bill.
10. Informal To overwhelm with hilarity, pleasure, or admiration: The outstanding finale killed the audience.
11. Sports a. To hit (a ball) with great force.
b. To hit (a ball) with such force as to make a return impossible, especially in a racquet game.
âintr. 1. To cause death or extinction; be fatal.
3. Informal To make such a strong impression as to overcome: dress to kill. *
n. 1. The act of killing.
2a. An animal killed, especially in hunting.
b. A person killed or to be killed: "Infantrymen⌠had seen too many kills suddenly get up and run away or shoot at them as they approached" (Nelson DeMille).
c. An enemy aircraft, vessel, or missile that has been attacked and destroyed.
â phrasal verb: kill off To destroy in such large numbers as to render extinct.
â idiom: in at (or on) the kill Present at the moment of triumph.
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n. 1. One that kills: a disease that was a killer of thou-sands; a killer of new ideas.
2. Slang Something that is extremely difficult to deal with or withstand: an exam that was a real killer.
adj. 1. Causing death or destruction: killer floods.
2. Slang Having impressive or effective power or impact; formidable: had a killer smile; made killer profits.
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What you can see with all the orange text is this idea of overwhelm. That it is a figurative expression of causing someone to have a death-like reaction. Think swooning, fainting, not being able to think, getting tongue tied. Basically any reaction where the extremity of it causes someone to cease normal functioning for a moment. And it happens because they just can't stand how attractive the Lady Killer is.
While I don't see anything directly, I would suspect (in other words take it with a whole bucket of salt) that the idea of pleasure predates the idea of admiration. Part of the problem is that, until very recently, the first time you'll see anything in text is usually long after it has been in use verbally. However, there is an extremely old connection between sex and violence in the English language.
There is potential textual evidence as early as 1503 that "fuck" was in use in English and the most likely explanation of where it comes from is the PIE root *peug meaning "to prick, pierce, or sting." It's not entirely clear where it makes the jump in the middle. There are a lot of similar words in other Germanic languages but there's also the influence of Norman French on us, where we would get the word derived from Latin pugnare "to fight." Either way, words like Pug, Pugnacious, Punch, Pugilist, Point, etc. are most likely related to Fuck. It doesn't just mean to have sex with, it means to have sex with in a, at least somewhat, violent manner.
So, the admiration sense of kill, almost always used in a context of the killer having power, and in the sub-context of appearance, almost always libidinous power, may reflect the idea of lust, not just looking good. Because the future or past killing in reference is potentially going to be something along the lines of a "little death," orgasm. They are a Lady Killer because they are good in bed.
But, if it is merely admiration, it is still certainly admiration for the Lady Killer's libidinous power to get ladies to fall at their feet by one means or another.