looking for Leiden-like City for Size, beauty, and ease biking around the City

The first place that comes to mind is Ghent.

Copenhagen is significantly larger, but extremely bike friendly and very easy to navigate. I'd be willing to wager, though, that some of Denmark's smaller cities would bit the bill, though I can't speak from direct experience.

Padova (Padua), a university city of a comparable size. Bologna, although a lot bigger, is also bicycle friendly.
And nowadays a lot of German cities, although they still have the irritating habit to send cyclists to the pavement.
I forgot to mention Hradec Kralove, in the Czech Republic, with an automated bicycle parking in front of the railway station.

normprice, it would be EXTREMELY helpful if you could add your home city and country to your profile. We can't tell anything about your location.

thank you for your posts so far. My son and I are now in Leiden, NL and are planing to start next leg on Saturday.

normprice, you don't say how far you are willing to go, but maybe Vienna should be on your list. We have excellent cycling infrastructure. The Ringstrasse is a delight by bike, the Prater is a nice place to ride, and the long distance cycle routes along the Danube are great for short rides outside the city. You can take your bike on the train to Melk, visit the abbey and then bike down the Danube to Krems on the Donauradweg and return to Vienna by train.

If you don't bring your own bikes, there is a city bike program and also firms that rent bikes by the day. These folks have decent equipment:
https://www.pedalpower.at/z_english/bikerental.php

There will be more tourists here, but it's manageable.