After Belgium would you recommend Den Haag or Rotterdam

Den Haag has great historic architecture with the Binnenhof, once a medieval castle, now serving as the seat of the Netherlands’ Parliament Nearby is the 1885 shopping arcade called “The Passage.” Back in a day, this was the largest shopping center in the Netherlands.
You can stay in Den Hague and take the scenic Tram Line 1 ride to Delft- which is just 5 miles away.
This is the slow, scenic route so it takes about 30 minutes.
Leiden, a university town of canals and medieval architecture of its renowned university that’s been called the “Oxford of the Netherlands,” is just 20-train-minutes away from The Hague.
Leiden’s home to a world-class Archeology Museum — the “ National Museum of Antiquities”— centuries-old botanical gardens, and an American Pilgrim Museum in one of the oldest wood buildings in Leiden. Leiden was the departure port from where the Mayflower sailed with the pilgrims who landed at Plymouth Rock in 1620.

Rotterdam was bombed heavily at the start of WW2. There are some parts that escaped the bombs and the fire that followed, the Delfshaven neighborhood for instance, but I wouldn’t describe Rotterdam as a quaint town. There is however lots of interesting modern architecture to see there. For instance the qubic houses and the Markthal.
On the other hand I wouldn’t describe The Hague as a quaint town either. The Binnenhof, the seat of our parliament, is undergoing major renovations and will be inaccessible for the foreseeable future. Perhaps even for the next decade. If you like Indonesian food though, The Hague is the place to be.
It’s a shame you can’t find a place to stay in Delft, because from what you describe it seems to fit the bill perfectly. Perhaps you can reconsider and stay in a hotel?

Rotterdam was definitely not quaint or charming, but we loved taking the water taxi to see the windmills from there. Delft had charm, but we were there on Kings Day this past spring, so it was party charm!

Month and year of trip?

Den Haag is perfect for your interests. Like Amsterdam, there are plenty of places within an hour on the train, too. Have you considered Leiden as a base? The fact is that the two main art museums in Den Haag are superior to anything in plenty nice Leiden.

Please remember that AirBnb frequently harms local residents more than it allegedly helps them. Even my wealthy commuter town (to NYC) bans them.

Mauritshuis in The Hague has one of the best collections of Flemish and Dutch paintings from the 15th to the 17th century. (you probably knew that)

https://www.mauritshuis.nl/en/

We chose to stay in Utrecht for our week in the Netherlands, and we'd do the same in future. It's much less hectic than Rotterdam and Amsterdam (and The Hague, too, actually) yet is a great transit hub for visiting other Dutch places.

I will admit to finding Delft to be unenjoyable. Each to one's own.

Thank you all very much for your thoughts. I've decided to stay in Den Haag and will add Museum Panorama Mesdag as a site to visit along with Mauritshuis and the Escher Palace. We will consider a side trip to Leiden as what American can pass up a visit to a museum dedicated to Pilgrims? I'm looking forward to this trip to Holland, although it will be brief.

For Americans interested in the Pilgrim Fathers is also worth visiting Delfshaven in Rotterdam. It’s the harbour where they set sail to England for joining the voyage with the Mayflower. The Pilgrim Fathers’Church is still in use, you can contact them for more info. For the rest is Delfshaven not so of interest, but the tiny historic harbour there is. https://www.pelgrimvaderskerk.nl/contact/

For more info about the Pilgrims in Leiden you can contact the local TI. https://www.visitleiden.nl/en/plan-your-visit/tourist-information and https://www.visitleiden.nl/en/highlights/pilgrims

I've decided to stay in Den Haag

Great choice. If the weather is nice while there, the seaside in Scheveningen is an interesting tram ride from the center of Den Haag.

We enjoyed our stay in Den Hagg this April at the Staybridge Suites Hotel. Friendly and helpful staff, nice breakfast and good location. Minutes form the Maruitshuis. Enjoy your visit. .

In case no one already mentioned this, from Den Haag, you can very easily and quickly take a tram (or the train( to Delft in about 15 minutes and spend the day or part of it in Delft since you wanted to go there.

At the risk of sounding like a clueless American tourist (which I am in some respects) is Den Haag The Hague? We will be spending just 2 nights in Delft but read about the tram ride into The Hague and would like to investigate that more.

is Den Haag The Hague?

Yes

Anyone interested in the Pilgrims can check out the Pelgrim brewery -- correct spelling -- for a congenial if modern brewpub and cafe.

https://www.greeka.com/cyclades/mykonos/car-bus/#:~:text=In%20brief%2C%20KTEL%20buses%20are,touristy%20spots%20on%20the%20island.

I've been to both and highly recommend Den Haag. Easy to get to Delft on the tram or train.

My wife is Dutch. She managed to convince me to spend our two week honeymoon in Utrecht in 2018. Great decision. Accessible by train to so many great towns and cities. Like others I fell in love with Utrecht. More recently we spent a week in Delft and loved that too. Close to so many attractions as others have noted, such as Leiden and Den Haag, both trip highlights. Next week we will spend one week in Alkmaar which looks very promising and will allow us to explore areas that I haven't been to before, plus return visits to Amsterdam. Then a week in Arnhem, putting us closer to other parts of the country. Amsterdam is great for visiting on day trips, but I highly recommend places such as Utrecht, Delft, Leiden and Den Haag for overnights.

We will consider a side trip to Leiden as what American can pass up a visit to a museum dedicated to Pilgrims?

For the museum, try to make reservations ahead. The Museum does not see heavy traffic, but it is small, just a tiny house, so 6-8 people fill it up

The Hague is fine as a base, others mentioned Utrecht, while that works, it is a bit out of the way for Rotterdam, Delft, and The Hague, area, if that is your focus. Utrecht does work great if you are looking at Amsterdam and cities to the East (Arnhem for example) as well.

My husband and I spent two of our three weeks based out of The Hague. We loved it there.