You say 'drive' but I assume you are using public transport in and around Leiden? Having a car in western Netherlands is not always practical... high parking fees and lots of traffic jams.
There are direct trains from Schiphol airport to Leiden, probably only takes about 15 minutes. (See that www.ns.nl/en website.)
As said earlier, you don't need to set everything in stone right now. You can visit Amsterdam two days in a row ,or on day 3 and 5 like you say now... but it really doesn't matter. Let the weather be a deciding factor as well... for example Keukenhof is a lot more beautiful if the sun shines and probably not much fun if it rains. Check weather forecasts every day and for the next day. Also install the apps Buienalarm, and Buienradar, they predict rain for the next 2 hours on a very local basis and it's quite accurate. This helps you know when to take shelter in a nice cafe or a museum, or when it's safe to stroll past beautiful canals in any of the cities you want to visit. :-)
Personally I really dislike Brussels. I think Ghent and Bruges are much nicer (could be visited from Rotterdam), or Antwerp which is even closer to Rotterdam.
I also heard Liege is nice (though in a rough, industrial way), this is not so far from Maastricht.
In Aachen don't miss the impressive cathedral.
And like I said earlier I think it's easier to find guided tours once you are in Delft or in The Hague. Not all the way from Leiden. But then again there might be tours that offer a combination, I don't know... ask the Leiden VVV tourist info bureau. There might be private tour operators offering this but it will be a lot more expensive than simply visiting those cities independently (which is perfectly doable and easy).
Our biggest city Amsterdam does not even have a million inhabitants (but 863,000). I think in the eyes of most foreigners, we only have a lot of 'villages'. :-)
My tips for Amsterdam are here: https://obertop.wordpress.com/travel-tips/amsterdam-tips/
If you want to visit Anne Frank House, or Van Gogh Museum (which has a David Hockney combi display this spring), get tickets in advance. I believe for Anne Frank tickets come out about 2 months in advance, and if it seems sold out for your dates, keep trying. They release more tickets in small batches over time, sometimes even on the same day of visiting.
Other museums are fine without advance tickets. Rijksmuseum may have a short line but doable. The cloakroom can get a long line; use a locker instead (if you don't want to carry your coat and bag into the museum). The lockers require a 0.50 or 1 euro coin (not sure which). Rijksmuseum's shop has some surprisingly nice items for sale. The museum itself is huge (plenty for 4 hours or more) so choose which parts you want to see, if you go there.