It's intended as a form of wordplay. As the comments note, to sleep one's way to the top is an idiom that means the trading of sexual favours for advancement of some sort. Although it can be applied to both sexes, it is usually an insult leveled at women who, correctly or not, are perceived as being unqualified for the positions they hold. (The stereotypical unqualified secretary, for instance.)
Given this context, the phrase can be seen as empowering for women who nap to increase their productivity. Stated plainly, it would be something like this:
You know how they say you only got where you are because you slept your way there? Well, you know they're wrong in terms of sexual favours, but make it a literal reality. Sleep more. Be more productive with a nap. Turn the insult into personal betterment.
The context includes "workplaces men have designed", so it's fairly clear that Huffington intends for her listeners to pick up the double meaning. She's basically saying, "Take that insult, embrace it, and make it backfire."
In some ways, it can be seen as analogous to the old joke:
Three men walked into a bar. Two were fine, but one had to have stitches.
You expect one thing from the sentence, and then when the full context is given, there's a shift in your perceptions. Huffington is doing the same thing with the phrase "sleep your way to the top".
The landing is simply the part of the floor next to the top or bottom of a flight of stairs. Landings are usually separated from rooms by doors and walls. In homes, where landings do not need to be very wide because people are few, doors to bedrooms are often within arm's reach of a landing.
Where a landing ends and (the rest of) a hallway begins is not that precise. Landings do not have to be a part of true hallways (or 'corridors') but can be surrounded on two or three sides by doors or walls.
Best Answer
Skimming (or skimming off the top) is a type of fraud. From Wikipedia: