If the prices are even in the same ballpark/universe and you are choosing between those two, go with LondonHouse. I know that thewit has its fans, but while it's kind of interesting and modern-ish/artsy-ish (think W, sort of), the LondonHouse is a truly high-end property.
Personally, I consider the Drake and WA to be in walking distance but my idea of walking distance is an outlier. By the combination of price, location, and offering, I've never found myself staying at the Drake (though I do like to just pop into the property for historical appeal). The WA used to be my go-to for high-end, but the LH has honestly changed that.
If you want to be right "in the thick of things" per a more normal definition than my love of 30-minute walks, I would stick with the LondonHouse, wit, PH, or Hilton - or one of the lower-tier properties in the immediate vicinity.