agree with york castle museum, has a fully recreated victorian and edwardian street - really interesting..lots of shop windows to look in from that time, toys etc.
or the dungeons if they are brave.
city centre is very compact, all walkable. the shambles (narrow street) fun to wander around and some interesting shops - maybe not so appealing for the DC though!
can't help with places ot stay, we always go for the day on the train.