How can I kill Beitild without anyone seeing?

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.

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.

I managed to kill her in an entirely conventional way and got away with it, but not in her house but outside at the smelter. I just waited until night, when most of the other citizens where away and whacked her while no guard was looking.

You might have to try that several times, there are guards making their rounds and you don't want to be seen by them.