A CRJ Captain elected to taxi to his gate after landing despite a nosewheel steering failure. While doing so he failed to hold short of an intervening runway as cleared.

Narrative

We landed normally on Runway 22L in JFK following a manual landing gear extension event. During the landing rollout we received a 'Steering INOP' caution; which was anticipated by the QRH checklist and noted. Winds were light and ground dry and I had excellent directional control with small differential brake and thrust inputs. I carefully exited at the end of the runway (Taxiway J was closed for construction) on Taxiway Z. The entire southeast end of Runway 13R/31L was closed east of Runway 22R/4L due to construction. We completed the after landing checklist and were instructed by Tower to stay with him and hold short of 13R. I decided I could safely; if slowly; maneuver the airplane towards the terminal ramp area so as not to cause disruption to traffic landing and taxiing behind us.After completing the after landing checklist; the First Officer briefly contacted (with my consent) JFK Operations to verify gate assignment and notify them we would require a tow team to bring us in to the gate after arriving on the ramp. I continued to slowly taxi the aircraft on what I thought was a continuation of Taxiway Z (there were construction barriers all around us and Z was closed north of the end of Runway 13R/31L). When the First Officer was done with his operations radio call; the Tower reminded us of our hold short clearance as we realized I had inadvertently taxied onto the closed section (but used for taxi) of Runway 13R/31L. The airport was departing traffic on 13R at the time. We were instructed to continue and make a right turn on Taxiway Y and contact Ground Control without further comment. We were then given clearance to the terminal ramp; arriving without incident and were towed into our parking space.This unintentional clearance deviation could have been prevented had I reversed my decision to continue taxi without the nosewheel steering system. My concern for causing operational distress from stopping just clear of the runway overrode a more conservative approach to the situation. Also; had I insisted the First Officer not contact Operations for gate confirmation (a common practice); we would possibly not have missed the hold-short. My concentration on taxiing the aircraft with inoperative steering reduced my situational awareness of our route; although it was assigned through a construction area looked like a taxi-only surface. I do not believe; in this instance; that safety was truly compromised. In the future; should I have a similar situation; I would more strongly consider stopping for tow and have both pilots monitor taxi progress.