How can I kill Beitild without anyone seeing?

So I'm on the first part of the Dark Brotherhood questline, and I have to kill Beitild in Dawnstar. I've done it before on a previous character, but I can't for the life of me seem to get it to work here.

I'm entering her house, under stealth, no one sees me, and I use my bow to one-shot her in the back whilst she is sleeping.

It seems that for some reason, the mage who lives in the house NEXT DOOR is a witness to my crime, despite him being in, you know, another house, and he summons the guards and tries to kill me.

Any advice?

Solution 1:

The easiest solution actually is just to tell her that you are there to slay her.

She will then scream something like "Come and get me!" and attack you.

After that you can slay her without getting bothered by the guards.

I do not know if this option appears all the time though or if it's connected to i.e. your Speechcraft skill or something else.

Solution 2:

My preferred method of dealing with the... stickier public assassination missions is a high level Frenzy poison and the Poisoner perk from the Pickpocket skill line.

Should get you off scot free every time.