Warkworth Castle (in Northumberland) has a beautiful and well-preserved keep, full of narrow staircases and secret passages. Not that these were ever used for the nefarious purposes of which I think when I daydream of castles with secret passages, mind you - they seem to have been built primarily for allowing servants to serve food and drink without wandering through the main rooms. One clear example of this being the stairs from the wine cellar directly up into the great hall - very practical!
We arrived just after a coach-load of small children, which for a moment looked like it could have been very bad timing. Fortunately the kids were being carefully herded around by teachers and parents in quite big groups, so we didn't find ourselves tripping over them all the time. And on the plus side, there were several costumed volunteers on hand to work with the school group, which added to the atmosphere more generally.
There's also a hermitage, accessible only by boat - the boats only run on certain days, and we unfortunately missed it, so it's worth checking to see if you can time your visit to coincide with the boats.