How to kill Grelod the Kind secretly?

Its not necessary as you do not receive any bounty for the murder even if you just walk up to her and stab her in the face. She's well known for being cruel to children so everyone sort of looks the other way.

I poisoned her to death and Michel still reacted in the same way, so I think its most likely scripted.

I had already done the thieves guild by this point and

had the nightingale's special powers,

and so had the ability of invisibility (I assume any other spell or ability would work as well). I snuck in, was unnoticed, killed her and turned invisible before i was 'noticed' and left without the eye icon ever opening at all.

That's the best I think you can get - all the kids ran in and the help soon did also, but no one ever ran from me as if they were afraid or anything. As other people had said, it's unimportant and completely irrelevant, but it may be of note that this is how I did it. (I also never had anyone note that how I killed her was messy or anything either.)

The help noticed me even when I was invisible, snuck into her room, closed the door and arrow'd her in the mouth. She came in and started running around like a nark. I think it's just scripted guys, no biggy anyway.

Invisibility is the way to go, but you must remember you're invisible, not untouchable. Also, you must kill her in one hit. I did this by (no magic) sneaking into her room. It doesn't matter if you're seen on the way in, close the doors to her room, backstab for 30x Damage (shrouded gloves from the brotherhood + sneaking perk) and finally, 1 stolen bottle of invisibility that I drank as I sat in the corner while everybody ran into the room. I then simply walked past them without touching them and out the door. The script that caused the lady to run screaming that she yields and cannot best me never started.